Grossman Health 2009-02-14

Chapter 7

THE HUMAN CAPITAL MODEL*

MICHAEL GROSSMAN

City University of New York Graduate School and National Bureau of Economic Research

Contents

Abstract 3481. Introduction 3492. Basic model 352

2.1. Assumptions 3522.2. Equilibrium conditions 3542.3. Optimal length of life 3562.4. "Bang-bang" equilibrium 3632.5. Special cases 366

3. Pure investment model 3673.1. Depreciation rate effects 3693.2. Market and nonmarket efficiency 371

4. Pure consumption model 3745. Empirical testing 377

5.1. Structure and reduced form 3785.2. Data and results 381

6. Extensions 3836.1. Empirical extensions with cross-sectional data 3836.2. Empirical extensions with longitudinal data 3886.3. Theoretical extensions 392

7. Health and schooling 3958. Conclusions 404References 405

'I am indebted to Robert Kaestner, Sara Markowitz, Tomas Philipson, and Walter Ried for helpful comments.

Ch. 7: The Human Capital Model

1. Introduction

Almost three decades have elapsed since I published my National Bureau of EconomicResearch monograph [Grossman (1972b)] and Journal of Political Economy paper[Grossman (1972a)] dealing with a theoretical and empirical investigation of the de-mand for the commodity "good health." l My work was motivated by the fundamentaldifference between health as an output and medical care as one of a number of inputsinto the production of health and by the equally important difference between healthcapital and other forms of human capital. According to traditional demand theory, eachconsumer has a utility or preference function that allows him or her to rank alternativecombinations of goods and services purchased in the market. Consumers are assumedto select that combination that maximizes their utility function subject to an incomeor resource constraint: namely, outlays on goods and services cannot exceed income.While this theory provides a satisfactory explanation of the demand for many goodsand services, students of medical economics have long realized that what consumersdemand when they purchase medical services are not these services per se but ratherbetter health. Indeed, as early as 1789, Bentham included relief of pain as one of fifteen"simple pleasures" which exhausted the list of basic arguments in one's utility function[Bentham (1931)]. The distinction between health as an output or an object of choiceand medical care as an input had not, however, been exploited in the theoretical andempirical literature prior to 1972.

My approach to the demand for health has been labeled as the human capital modelin much of the literature on health economics because it draws heavily on human capitaltheory [Becker (1964, 1967), Ben-Porath (1967), Mincer (1974)]. According to humancapital theory, increases in a person's stock of knowledge or human capital raise hisproductivity in the market sector of the economy, where he produces money earnings,and in the nonmarket or household sector, where he produces commodities that enter hisutility function. To realize potential gains in productivity, individuals have an incentiveto invest in formal schooling and on-the-job training. The costs of these investmentsinclude direct outlays on market goods and the opportunity cost of the time that mustbe withdrawn from competing uses. This framework was used by Becker (1967) and byBen-Porath (1967) to develop models that determine the optimal quantity of investmentin human capital at any age. In addition, these models show how the optimal quantityvaries over the life cycle of an individual and among individuals of the same age.

Although Mushkin (1962), Becker (1964), and Fuchs (1966) had pointed out thathealth capital is one component of the stock of human capital, I was the first personto construct a model of the demand for health capital itself. If increases in the stock ofhealth simply increased wage rates, my undertaking would not have been necessary, for

1 My monograph was the final publication in the Occasional Paper Series of the National Bureau of Eco-nomic Research. It is somewhat ironic that the publication of a study dealing with the demand for healthmarked the death of the series.

1. Introduction

Almost three decades have elapsed since I published my National Bureau of EconomicResearch monograph [Grossman (1972b)] and Journal of Political Economy paper[Grossman (1972a)] dealing with a theoretical and empirical investigation of the de-mand for the commodity "good health." l My work was motivated by the fundamentaldifference between health as an output and medical care as one of a number of inputsinto the production of health and by the equally important difference between healthcapital and other forms of human capital. According to traditional demand theory, eachconsumer has a utility or preference function that allows him or her to rank alternativecombinations of goods and services purchased in the market. Consumers are assumedto select that combination that maximizes their utility function subject to an incomeor resource constraint: namely, outlays on goods and services cannot exceed income.While this theory provides a satisfactory explanation of the demand for many goodsand services, students of medical economics have long realized that what consumersdemand when they purchase medical services are not these services per se but ratherbetter health. Indeed, as early as 1789, Bentham included relief of pain as one of fifteen"simple pleasures" which exhausted the list of basic arguments in one's utility function[Bentham (1931)]. The distinction between health as an output or an object of choiceand medical care as an input had not, however, been exploited in the theoretical andempirical literature prior to 1972.

My approach to the demand for health has been labeled as the human capital modelin much of the literature on health economics because it draws heavily on human capitaltheory [Becker (1964, 1967), Ben-Porath (1967), Mincer (1974)]. According to humancapital theory, increases in a person's stock of knowledge or human capital raise hisproductivity in the market sector of the economy, where he produces money earnings,and in the nonmarket or household sector, where he produces commodities that enter hisutility function. To realize potential gains in productivity, individuals have an incentiveto invest in formal schooling and on-the-job training. The costs of these investmentsinclude direct outlays on market goods and the opportunity cost of the time that mustbe withdrawn from competing uses. This framework was used by Becker (1967) and byBen-Porath (1967) to develop models that determine the optimal quantity of investmentin human capital at any age. In addition, these models show how the optimal quantityvaries over the life cycle of an individual and among individuals of the same age.

Although Mushkin (1962), Becker (1964), and Fuchs (1966) had pointed out thathealth capital is one component of the stock of human capital, I was the first personto construct a model of the demand for health capital itself. If increases in the stock ofhealth simply increased wage rates, my undertaking would not have been necessary, for

1 My monograph was the final publication in the Occasional Paper Series of the National Bureau of Eco-nomic Research. It is somewhat ironic that the publication of a study dealing with the demand for healthmarked the death of the series.

one could simply have applied Becker's and Ben-Porath's models to study the decisionto invest in health. I argued, however, that health capital differs from other forms of hu-man capital. In particular, I argued that a person's stock of knowledge affects his marketand nonmarket productivity, while his stock of health determines the total amount oftime he can spend producing money earnings and commodities.

My approach uses the household production function model of consumer behavior[Becker (1965), Lancaster (1966), Michael and Becker (1973)] to account for the gapbetween health as an output and medical care as one of many inputs into its production.This model draws a sharp distinction between fundamental objects of choice - calledcommodities - that enter the utility function and market goods and services. Thesecommodities are Bentham's (1931) pleasures that exhaust the basic arguments in theutility function. Consumers produce commodities with inputs of market goods and ser-vices and their own time. For example, they use sporting equipment and their own timeto produce recreation, traveling time and transportation services to produce visits, andpart of their Sundays and church services to produce "peace of mind." The concept ofa household production function is perfectly analogous to a firm production function.Each relates a specific output or a vector of outputs to a set of inputs. Since goods andservices are inputs into the production of commodities, the demand for these goods andservices is a derived demand for a factor of production. That is, the demand for medicalcare and other health inputs is derived from the basic demand for health.

There is an important link between the household production theory of consumerbehavior and the theory of investment in human capital. Consumers as investors in theirhuman capital produce these investments with inputs of their own time, books, teachers'services, and computers. Thus, some of the outputs of household production directlyenter the utility function, while other outputs determine earnings or wealth in a lifecycle context. Health, on the other hand, does both.

In my model, health - defined broadly to include longevity and illness-free days in agiven year - is both demanded and produced by consumers. Health is a choice variablebecause it is a source of utility (satisfaction) and because it determines income or wealthlevels. That is, health is demanded by consumers for two reasons. As a consumptioncommodity, it directly enters their preference functions, or, put differently, sick days area source of disutility. As an investment commodity, it determines the total amount oftime available for market and nonmarket activities. In other words, an increase in thestock of health reduces the amount of time lost from these activities, and the monetaryvalue of this reduction is an index of the return to an investment in health.

Since health capital is one component of human capital, a person inherits an initialstock of health that depreciates with age - at an increasing rate at least after some stagein the life cycle - and can be increased by investment. Death occurs when the stockfalls below a certain level, and one of the novel features of the model is that individuals"choose" their length of life. Gross investments are produced by household productionfunctions that relate an output of health to such choice variables or health inputs asmedical care utilization, diet, exercise, cigarette smoking, and alcohol consumption. Inaddition, the production function is affected by the efficiency or productivity of a given

350 M. Grossman

one could simply have applied Becker's and Ben-Porath's models to study the decisionto invest in health. I argued, however, that health capital differs from other forms of hu-man capital. In particular, I argued that a person's stock of knowledge affects his marketand nonmarket productivity, while his stock of health determines the total amount oftime he can spend producing money earnings and commodities.

My approach uses the household production function model of consumer behavior[Becker (1965), Lancaster (1966), Michael and Becker (1973)] to account for the gapbetween health as an output and medical care as one of many inputs into its production.This model draws a sharp distinction between fundamental objects of choice - calledcommodities - that enter the utility function and market goods and services. Thesecommodities are Bentham's (1931) pleasures that exhaust the basic arguments in theutility function. Consumers produce commodities with inputs of market goods and ser-vices and their own time. For example, they use sporting equipment and their own timeto produce recreation, traveling time and transportation services to produce visits, andpart of their Sundays and church services to produce "peace of mind." The concept ofa household production function is perfectly analogous to a firm production function.Each relates a specific output or a vector of outputs to a set of inputs. Since goods andservices are inputs into the production of commodities, the demand for these goods andservices is a derived demand for a factor of production. That is, the demand for medicalcare and other health inputs is derived from the basic demand for health.

There is an important link between the household production theory of consumerbehavior and the theory of investment in human capital. Consumers as investors in theirhuman capital produce these investments with inputs of their own time, books, teachers'services, and computers. Thus, some of the outputs of household production directlyenter the utility function, while other outputs determine earnings or wealth in a lifecycle context. Health, on the other hand, does both.

In my model, health - defined broadly to include longevity and illness-free days in agiven year - is both demanded and produced by consumers. Health is a choice variablebecause it is a source of utility (satisfaction) and because it determines income or wealthlevels. That is, health is demanded by consumers for two reasons. As a consumptioncommodity, it directly enters their preference functions, or, put differently, sick days area source of disutility. As an investment commodity, it determines the total amount oftime available for market and nonmarket activities. In other words, an increase in thestock of health reduces the amount of time lost from these activities, and the monetaryvalue of this reduction is an index of the return to an investment in health.

Since health capital is one component of human capital, a person inherits an initialstock of health that depreciates with age - at an increasing rate at least after some stagein the life cycle - and can be increased by investment. Death occurs when the stockfalls below a certain level, and one of the novel features of the model is that individuals"choose" their length of life. Gross investments are produced by household productionfunctions that relate an output of health to such choice variables or health inputs asmedical care utilization, diet, exercise, cigarette smoking, and alcohol consumption. Inaddition, the production function is affected by the efficiency or productivity of a given

350 M. Grossman

consumer as reflected by his or her personal characteristics. Efficiency is defined as theamount of health obtained from a given amount of health inputs. For example, years offormal schooling completed plays a large role in this context.

Since the most fundamental law in economics is the law of the downward slopingdemand function, the quantity of health demanded should be negatively correlated withits "shadow price." I stress that the shadow price of health depends on many other vari-ables besides the price of medical care. Shifts in these variables alter the optimal amountof health and also alter the derived demand for gross investment and for health inputs.I show that the shadow price of health rises with age if the rate of depreciation onthe stock of health rises over the life cycle and falls with education (years of formalschooling completed) if more educated people are more efficient producers of health.I emphasize the result that, under certain conditions, an increase in the shadow pricemay simultaneously reduce the quantity of health demanded and increase the quantitiesof health inputs demanded.

The task in this paper is to outline my 1972 model of the demand for health, to discussthe theoretical predictions it contains, to review theoretical extensions of the model, andto survey empirical research that tests the predictions made by the model or studiescausality between years of formal schooling completed and good health. I outline mymodel in Section 2 of this paper. I include a new interpretation of the condition for death,which is motivated in part by analyses by Ehrlich and Chuma (1990) and by Ried (1996,1998). I also address a fundamental criticism of my framework raised by Ehrlich andChuma involving an indeterminacy problem with regard to optimal investment in health.I summarize my pure investment model in Section 3, my pure consumption model inSection 4, and my empirical testing of the model in Section 5. While I emphasize myown contributions in these three sections, I do treat closely related developments thatfollowed my 1972 publications. I keep derivations to a minimum because these can befound in Grossman (1972a, 1972b).2 In Section 6 I focus on theoretical and empiri-cal extensions and criticisms, other than those raised by Ehrlich and Chuma and byRied.

I conclude in Section 7 with a discussion of studies that investigate alternative ex-planations of the positive relationship between years of formal schooling completedand alternative measures of adult health. While not all this literature is grounded indemand for health models, it is natural to address it in a paper of this nature becauseit essentially deals with complementary relationships between the two most importantcomponents of the stock of human capital. Currently, we still lack comprehensive the-oretical models in which the stocks of health and knowledge are determined simulta-neously. I am somewhat disappointed that my 1982 plea for the development of thesemodels has gone unanswered [Grossman (1982)]. The rich empirical literature treat-ing interactions between schooling and health underscores the potential payoffs to thisundertaking.

2 Grossman (1972b) is out of pnnt but available in most libraries.

2. Basic model

2.1. Assumptions

Let the intertemporal utility function of a typical consumer be

U = U(tHt, Z), t = 0, 1 .. , n, (1)

where Ht is the stock of health at age t or in time period t, ,t is the service flow per unitstock, ht = Pt Ht is total consumption of "health services," and Zt is consumption ofanother commodity. The stock of health in the initial period (H0) is given, but the stockof health at any other age is endogenous. The length of life as of the planning date (n)also is endogenous. In particular, death takes place when Ht < Hmin. Therefore, lengthof life is determined by the quantities of health capital that maximize utility subject toproduction and resource constraints.

By definition, net investment in the stock of health equals gross investment minusdepreciation:

Ht+ - Ht = It - tHt, (2)

where It is gross investment and 8t is the rate of depreciation during the tth period(0 < t < 1). The rates of depreciation are exogenous but depend on age. Consumersproduce gross investment in health and the other commodities in the utility functionaccording to a set of household production functions:

It = It(Mt, THt; E), (3)

Zt = Zt (Xt, Tt; E). (4)

In these equations Mt is a vector of inputs (goods) purchased in the market thatcontribute to gross investment in health, Xt is a similar vector of goods inputs that con-tribute to the production of Zt, THt and Tt are time inputs, and E is the consumer'sstock of knowledge or human capital exclusive of health capital. This latter stock isassumed to be exogenous or predetermined. The semicolon before it highlights the dif-ference between this variable and the endogenous goods and time inputs. In effect, I amexamining the consumer's behavior after he has acquired the optimal stock of this cap-ital.3 Following Michael (1972, 1973) and Michael and Becker (1973), I assume thatan increase in knowledge capital raises the efficiency of the production process in thenonmarket or household sector, just as an increase in technology raises the efficiency ofthe production process in the market sector. I also assume that all production functionsare linear homogeneous in the endogenous market goods and own time inputs.

3 Equations (3) and (4) assume that E does not vary over the life cycle. In Grossman (1972b, pp. 28-30), Iconsider the impacts of exogenous variations in this stock with age.

352 M. Grossman

2. Basic model

2.1. Assumptions

Let the intertemporal utility function of a typical consumer be

U = U(tHt, Z), t = 0, 1 .. , n, (1)

where Ht is the stock of health at age t or in time period t, ,t is the service flow per unitstock, ht = Pt Ht is total consumption of "health services," and Zt is consumption ofanother commodity. The stock of health in the initial period (H0) is given, but the stockof health at any other age is endogenous. The length of life as of the planning date (n)also is endogenous. In particular, death takes place when Ht < Hmin. Therefore, lengthof life is determined by the quantities of health capital that maximize utility subject toproduction and resource constraints.

By definition, net investment in the stock of health equals gross investment minusdepreciation:

Ht+ - Ht = It - tHt, (2)

where It is gross investment and 8t is the rate of depreciation during the tth period(0 < t < 1). The rates of depreciation are exogenous but depend on age. Consumersproduce gross investment in health and the other commodities in the utility functionaccording to a set of household production functions:

It = It(Mt, THt; E), (3)

Zt = Zt (Xt, Tt; E). (4)

In these equations Mt is a vector of inputs (goods) purchased in the market thatcontribute to gross investment in health, Xt is a similar vector of goods inputs that con-tribute to the production of Zt, THt and Tt are time inputs, and E is the consumer'sstock of knowledge or human capital exclusive of health capital. This latter stock isassumed to be exogenous or predetermined. The semicolon before it highlights the dif-ference between this variable and the endogenous goods and time inputs. In effect, I amexamining the consumer's behavior after he has acquired the optimal stock of this cap-ital.3 Following Michael (1972, 1973) and Michael and Becker (1973), I assume thatan increase in knowledge capital raises the efficiency of the production process in thenonmarket or household sector, just as an increase in technology raises the efficiency ofthe production process in the market sector. I also assume that all production functionsare linear homogeneous in the endogenous market goods and own time inputs.

3 Equations (3) and (4) assume that E does not vary over the life cycle. In Grossman (1972b, pp. 28-30), Iconsider the impacts of exogenous variations in this stock with age.

352 M. Grossman

In much of my modeling, I treat the vectors of goods inputs, Mt and Xt, as scalarsand associate the market goods input in the gross investment production function withmedical care. Clearly this is an oversimplification because many other market goods andservices influence health. Examples include housing, diet, recreation, cigarette smoking,and excessive alcohol use. The latter two inputs have negative marginal products in theproduction of health. They are purchased because they are inputs into the production ofother commodities such as "smoking pleasure" that yield positive utility. In completingthe model I will rule out this and other types of joint production, although I considerjoint production in some detail in Grossman (1972b, pp. 74-83). I also will associate themarket goods input in the health production function with medical care, although thereader should keep in mind that the model would retain its structure if the primary healthinput purchased in the market was something other than medical care. This is importantbecause of evidence that medical care may be an unimportant determinant of health indeveloped countries [see Auster et al. (1969)] and because Zweifel and Breyer (1997)use the lack of a positive relationship between correlates of good health and medicalcare in micro data to criticize my approach.

Both market goods and own time are scarce resources. The goods budget constraintequates the present value of outlays on goods to the present value of earnings incomeover the life cycle plus initial assets (discounted property income):

PtMt+QtXt _ WtTWt+ (1 +r) t t +AO (5)

t=O ( r)=O

Here Pt and Qt are the prices of Mt and Xt, Wt is the hourly wage rate, TWt is hoursof work, Ao is initial assets, and r is the market rate of interest. The time constraintrequires that 2, the total amount of time available in any period, must be exhausted byall possible uses:

TWt + THt + Tt + TLt = Q2, (6)

where TLt is time lost from market and nonmarket activities due to illness and injury.Equation (6) modifies the time budget constraint in Becker's (1965) allocation of time

model. If sick time were not added to market and nonmarket time, total time would notbe exhausted by all possible uses. I assume that sick time is inversely related to thestock of health; that is aTLt/aHt < 0. If 2 is measured in hours (2 = 8,760 hours or365 days times 24 hours per day if the year is the relevant period) and if /t is defined asthe flow of healthy time per unit of Ht, ht equals the total number of healthy hours in agiven year. Then one can write

TLt = 2 - ht.

In much of my modeling, I treat the vectors of goods inputs, Mt and Xt, as scalarsand associate the market goods input in the gross investment production function withmedical care. Clearly this is an oversimplification because many other market goods andservices influence health. Examples include housing, diet, recreation, cigarette smoking,and excessive alcohol use. The latter two inputs have negative marginal products in theproduction of health. They are purchased because they are inputs into the production ofother commodities such as "smoking pleasure" that yield positive utility. In completingthe model I will rule out this and other types of joint production, although I considerjoint production in some detail in Grossman (1972b, pp. 74-83). I also will associate themarket goods input in the health production function with medical care, although thereader should keep in mind that the model would retain its structure if the primary healthinput purchased in the market was something other than medical care. This is importantbecause of evidence that medical care may be an unimportant determinant of health indeveloped countries [see Auster et al. (1969)] and because Zweifel and Breyer (1997)use the lack of a positive relationship between correlates of good health and medicalcare in micro data to criticize my approach.

Both market goods and own time are scarce resources. The goods budget constraintequates the present value of outlays on goods to the present value of earnings incomeover the life cycle plus initial assets (discounted property income):

PtMt+QtXt _ WtTWt+ (1 +r) t t +AO (5)

t=O ( r)=O

Here Pt and Qt are the prices of Mt and Xt, Wt is the hourly wage rate, TWt is hoursof work, Ao is initial assets, and r is the market rate of interest. The time constraintrequires that 2, the total amount of time available in any period, must be exhausted byall possible uses:

TWt + THt + Tt + TLt = Q2, (6)

where TLt is time lost from market and nonmarket activities due to illness and injury.Equation (6) modifies the time budget constraint in Becker's (1965) allocation of time

model. If sick time were not added to market and nonmarket time, total time would notbe exhausted by all possible uses. I assume that sick time is inversely related to thestock of health; that is aTLt/aHt < 0. If 2 is measured in hours (2 = 8,760 hours or365 days times 24 hours per day if the year is the relevant period) and if /t is defined asthe flow of healthy time per unit of Ht, ht equals the total number of healthy hours in agiven year. Then one can write

TLt = 2 - ht.

From now on, I assume that the variable ht in the utility function coincides with healthyhours.4

By substituting for hours of work (TWt) from Equation (6) into Equation (5), oneobtains the single "full wealth" constraint:

Pt Mt + QtXt + Wt(TLt + THt + T,) Wt

t=0 t=O= (1 +r)= + A

Full wealth, which is given by the right-hand side of Equation (8), equals initial assetsplus the discounted value of the earnings an individual would obtain if he spent all ofhis time at work. Part of this wealth is spent on market goods, part of it is spent onnonmarket production, and part of it is lost due to illness. The equilibrium quantitiesof Ht and Zt can now be found by maximizing the utility function given by Equation(1) subject to the constraints given by Equations (2), (3), and (8). Since the inheritedstock of health and the rates of depreciation are given, the optimal quantities of grossinvestment determine the optimal quantities of health capital.

2.2. Equilibrium conditions

First-order optimality conditions for gross investment in period - I are 5

7t I WtGt (1 -St)Wt+ 1 Gt++ +. ..

(1 + r) t - I (1 + r) t (1 + r) t + l

(1 - t) ' (1 - 6,_)W, G,,

(1 +r)f?

+ t + (I -St) (1 -n-l ) G,, (9)

Pt_ Wt,-inrt-j= -(10)

aIt-l/aMt-i I It- /aTHt- i '

The new symbols in these equations are: Uht = aU/aht, the marginal utility of healthytime; ., the marginal utility of wealth; Gt = aht/aHt = -(TLt/IaHt), the marginal

4 Clearly this is a simplification. No distinction is made between the quality and the quantity of healthy time.If the stock of health yielded other services besides healthy time, q5t would be a vector of service flows. Theseservices might or might not be perfect substitutes in the utility function.5 An increase in gross investment in period t - 1 increases the stock of health in all future periods. Theseincreases are equal to

aH, aHt+l t HI

354 M. Grossman

From now on, I assume that the variable ht in the utility function coincides with healthyhours.4

By substituting for hours of work (TWt) from Equation (6) into Equation (5), oneobtains the single "full wealth" constraint:

Pt Mt + QtXt + Wt(TLt + THt + T,) Wt

t=0 t=O= (1 +r)= + A

Full wealth, which is given by the right-hand side of Equation (8), equals initial assetsplus the discounted value of the earnings an individual would obtain if he spent all ofhis time at work. Part of this wealth is spent on market goods, part of it is spent onnonmarket production, and part of it is lost due to illness. The equilibrium quantitiesof Ht and Zt can now be found by maximizing the utility function given by Equation(1) subject to the constraints given by Equations (2), (3), and (8). Since the inheritedstock of health and the rates of depreciation are given, the optimal quantities of grossinvestment determine the optimal quantities of health capital.

2.2. Equilibrium conditions

First-order optimality conditions for gross investment in period - I are 5

7t I WtGt (1 -St)Wt+ 1 Gt++ +. ..

(1 + r) t - I (1 + r) t (1 + r) t + l

(1 - t) ' (1 - 6,_)W, G,,

(1 +r)f?

+ t + (I -St) (1 -n-l ) G,, (9)

Pt_ Wt,-inrt-j= -(10)

aIt-l/aMt-i I It- /aTHt- i '

The new symbols in these equations are: Uht = aU/aht, the marginal utility of healthytime; ., the marginal utility of wealth; Gt = aht/aHt = -(TLt/IaHt), the marginal

4 Clearly this is a simplification. No distinction is made between the quality and the quantity of healthy time.If the stock of health yielded other services besides healthy time, q5t would be a vector of service flows. Theseservices might or might not be perfect substitutes in the utility function.5 An increase in gross investment in period t - 1 increases the stock of health in all future periods. Theseincreases are equal to

aH, aHt+l t HI

354 M. Grossman

product of the stock of health in the production of healthy time; and 7rt- 1, the marginalcost of gross investment in health in period t - 1.

Equation (9) states that the present value of the marginal cost of gross investment inhealth in period t - 1 must equal the present value of marginal benefits. Discountedmarginal benefits at age t equal

Gt Wt Uh1( + r)t X

where Gt is the marginal product of health capital - the increase in the amount ofhealthy time caused by a one-unit increase in the stock of health. Two monetary magni-tudes are necessary to convert this marginal product into value terms because consumersdesire health for two reasons. The discounted wage rate measures the monetary valueof a one-unit increase in the total amount of time available for market and nonmarketactivities, and the term Uht/X measures the discounted monetary value of the increasein utility due to a one-unit increase in healthy time. Thus, the sum of these two termsmeasures the discounted marginal value to consumers of the output produced by healthcapital.

Condition (9) holds for any capital asset, not just for health capital. The marginalcost as of the current period, obtained by multiplying both sides of the equation by(1 + r) t - 1, must be equated to the discounted flows of marginal benefits in the future.This is true for the asset of health capital by labeling the marginal costs and benefitsof this particular asset in the appropriate manner. As I will show presently, most of theeffects of variations in exogenous variables can be traced out as shifting the marginalcosts and marginal benefits of the asset.

While Equation (9) determines the optimal amount of gross investment in periodt - 1, Equation (10) shows the condition for minimizing the cost of producing a givenquantity of gross investment. Total cost is minimized when the increase in gross invest-ment from spending an additional dollar on medical care equals the increase in total costfrom spending an additional dollar on time. Since the gross investment production func-tion is homogeneous of degree one in the two endogenous inputs and since the prices ofmedical care and time are independent of the level of these inputs, the average cost ofgross investment is constant and equal to the marginal cost.

To examine the forces that affect the demand for health and gross investment, it isuseful to convert Equation (9) into an equation that determines the optimal stock ofhealth in period t. If gross investment in period t is positive, a condition similar to (9)holds for its optimal value. From these two first-order conditions

Gt W (+ Uh, (1r)] =7rt 1 (r-fit1 +or), (11)

product of the stock of health in the production of healthy time; and 7rt- 1, the marginalcost of gross investment in health in period t - 1.

Equation (9) states that the present value of the marginal cost of gross investment inhealth in period t - 1 must equal the present value of marginal benefits. Discountedmarginal benefits at age t equal

Gt Wt Uh1( + r)t X

where Gt is the marginal product of health capital - the increase in the amount ofhealthy time caused by a one-unit increase in the stock of health. Two monetary magni-tudes are necessary to convert this marginal product into value terms because consumersdesire health for two reasons. The discounted wage rate measures the monetary valueof a one-unit increase in the total amount of time available for market and nonmarketactivities, and the term Uht/X measures the discounted monetary value of the increasein utility due to a one-unit increase in healthy time. Thus, the sum of these two termsmeasures the discounted marginal value to consumers of the output produced by healthcapital.

Condition (9) holds for any capital asset, not just for health capital. The marginalcost as of the current period, obtained by multiplying both sides of the equation by(1 + r) t - 1, must be equated to the discounted flows of marginal benefits in the future.This is true for the asset of health capital by labeling the marginal costs and benefitsof this particular asset in the appropriate manner. As I will show presently, most of theeffects of variations in exogenous variables can be traced out as shifting the marginalcosts and marginal benefits of the asset.

While Equation (9) determines the optimal amount of gross investment in periodt - 1, Equation (10) shows the condition for minimizing the cost of producing a givenquantity of gross investment. Total cost is minimized when the increase in gross invest-ment from spending an additional dollar on medical care equals the increase in total costfrom spending an additional dollar on time. Since the gross investment production func-tion is homogeneous of degree one in the two endogenous inputs and since the prices ofmedical care and time are independent of the level of these inputs, the average cost ofgross investment is constant and equal to the marginal cost.

To examine the forces that affect the demand for health and gross investment, it isuseful to convert Equation (9) into an equation that determines the optimal stock ofhealth in period t. If gross investment in period t is positive, a condition similar to (9)holds for its optimal value. From these two first-order conditions

Gt W (+ Uh, (1r)] =7rt 1 (r-fit1 +or), (11)

where it_ l is the percentage rate of change in marginal cost between period t - I andperiod t.6 Equation (11) implies that the undiscounted value of the marginal productof the optimal stock of health capital at any age must equal the supply price of cap-ital, rt-1 (r - t-1 + t). The latter contains interest, depreciation, and capital gainscomponents and may be interpreted as the rental price or user cost of health capital.

Equation (11) fully determines the optimal quantity at time t of a capital good thatcan be bought and sold in a perfect market. The stock of health capital, like the stockof knowledge capital, cannot be sold because it is imbedded in the investor. This meansthat gross investment cannot be nonnegative. Although sales of capital are ruled out,provided gross investment is positive, there exists a user cost of capital that in equilib-rium must equal the value of the marginal product of the stock. In Grossman (1972a,p. 230); (1972b, pp. 6-7), I provide an intuitive interpretation of this result by showingthat exchanges over time in the stock of health by an individual substitute for exchangesin the capital market.

2.3. Optimal length of life7

So far I have essentially reproduced the analysis of equilibrium conditions in my 1972National Bureau of Economic Research monograph and Journal of Political Economyarticle. A perceptive reader may have noted that an explicit condition determining lengthof life is absent. The discounted marginal benefits of an investment in health in period0 are summed from periods 1 through n, so that the consumer is alive in period n anddead in period n + 1.8 This means that Hn+ is equal to or less than Hmin, the deathstock, while Hn and Hr (t < n) exceed Hmin. But how do we know that the optimalquantities of the stock of health guarantee this outcome? Put differently, length of lifeis supposed to be an endogenous variable in the model, yet discounted income andexpenditure flows in the full wealth constraint and discounted marginal benefits in thefirst-order conditions appear to be summed over a fixed n.

I was bothered by the above while I was developing my model. As of the date ofits publication, I was not convinced that length of life was in fact being determinedby the model. There is a footnote in my Journal of Political Economy article [Gross-man (1972a, p. 228, footnote 7)] and in my National Bureau of Economic Researchmonograph [Grossman (1972b, p. 4, footnote 9)] in which I impose the constraints thatH,+l < Hmin and H0 > Hmin.9 Surely, it is wrong to impose these constraints in amaximization problem in which length of life is endogenous.

6 Equation (11) assumes Sti-1 --O.7 First-time readers of this chapter can skip Sections 2.3 and 2.4. The material in the remaining sectionsdoes not depend on them.8 Since the initial period is period 0, a consumer who is alive in year n and dead in year n + 1 lives for n + Iyears.9 Actually, I assert that I am assuming H. < Hmin. That is incorrect because H,n > Hmin if the consumer isalive in period n. The corrected footnote should read: "The constraints are imposed that Hn+l < Hmin andH, > Hmin."

356 M. Grossman

where it_ l is the percentage rate of change in marginal cost between period t - I andperiod t.6 Equation (11) implies that the undiscounted value of the marginal productof the optimal stock of health capital at any age must equal the supply price of cap-ital, rt-1 (r - t-1 + t). The latter contains interest, depreciation, and capital gainscomponents and may be interpreted as the rental price or user cost of health capital.

Equation (11) fully determines the optimal quantity at time t of a capital good thatcan be bought and sold in a perfect market. The stock of health capital, like the stockof knowledge capital, cannot be sold because it is imbedded in the investor. This meansthat gross investment cannot be nonnegative. Although sales of capital are ruled out,provided gross investment is positive, there exists a user cost of capital that in equilib-rium must equal the value of the marginal product of the stock. In Grossman (1972a,p. 230); (1972b, pp. 6-7), I provide an intuitive interpretation of this result by showingthat exchanges over time in the stock of health by an individual substitute for exchangesin the capital market.

2.3. Optimal length of life7

So far I have essentially reproduced the analysis of equilibrium conditions in my 1972National Bureau of Economic Research monograph and Journal of Political Economyarticle. A perceptive reader may have noted that an explicit condition determining lengthof life is absent. The discounted marginal benefits of an investment in health in period0 are summed from periods 1 through n, so that the consumer is alive in period n anddead in period n + 1.8 This means that Hn+ is equal to or less than Hmin, the deathstock, while Hn and Hr (t < n) exceed Hmin. But how do we know that the optimalquantities of the stock of health guarantee this outcome? Put differently, length of lifeis supposed to be an endogenous variable in the model, yet discounted income andexpenditure flows in the full wealth constraint and discounted marginal benefits in thefirst-order conditions appear to be summed over a fixed n.

I was bothered by the above while I was developing my model. As of the date ofits publication, I was not convinced that length of life was in fact being determinedby the model. There is a footnote in my Journal of Political Economy article [Gross-man (1972a, p. 228, footnote 7)] and in my National Bureau of Economic Researchmonograph [Grossman (1972b, p. 4, footnote 9)] in which I impose the constraints thatH,+l < Hmin and H0 > Hmin.9 Surely, it is wrong to impose these constraints in amaximization problem in which length of life is endogenous.

6 Equation (11) assumes Sti-1 --O.7 First-time readers of this chapter can skip Sections 2.3 and 2.4. The material in the remaining sectionsdoes not depend on them.8 Since the initial period is period 0, a consumer who is alive in year n and dead in year n + 1 lives for n + Iyears.9 Actually, I assert that I am assuming H. < Hmin. That is incorrect because H,n > Hmin if the consumer isalive in period n. The corrected footnote should read: "The constraints are imposed that Hn+l < Hmin andH, > Hmin."

356 M. Grossman

My publications on the demand for health were outgrowths of my 1970 ColumbiaUniversity Ph.D. dissertation. While I was writing my dissertation, my friend and fellowPh.D. candidate, Gilbert R. Ghez, pointed out that the determination of optimal length oflife could be viewed as an iterative process. I learned a great deal from him, and I oftenspent a long time working through the implications of his comments.l° It has taken mealmost thirty years to work through his comment on the iterative determination of lengthof life. I abandoned this effort many years ago but returned to it when I read Ried's(1996, 1998) reformulation of the selection of the optimal stock of health and lengthof life as a discrete time optimal control problem. Ried (1998, p. 389) writes: "Since[the problem] is a free terminal time problem, one may suspect that a condition for theoptimal length of the planning horizon is missing in the set of necessary conditions ....However, unlike the analogous continuous time problem, the discrete time version failsto provide such an equation. Rather, the optimal final period ... has to be determinedthrough the analysis of a sequence of fixed terminal time problems with the terminaltime varying over a plausible domain." This is the same observation that Ghez made.I offer a proof below. I do not rely on Ried's solution. Instead, I offer a much simplerproof which has a very different implication than the one offered by Ried.

A few preliminaries are in order. First, I assume that the rate of depreciation on thestock of health (,) rises with age. As we shall see in more detail later on, this impliesthat the optimal stock falls with age. Second, I assume that optimal gross investmentin health is positive except in the very last year of life. Third, I define Vt as Wt +(Uht/X)(l + r)t. Hence, Vt is the undiscounted marginal value of the output producedby health capital in period t. Finally, since the output produced by health capital hasa finite upper limit of 8,760 hours in a year, I assume that the marginal product of thestock of health (Gt) diminishes as the stock increases (G,/Hlat < 0).

Consider the maximization problem outlined in Section 2.1 except that the planninghorizon is exogenous. That is, an individual is alive in period n and dead in period n + 1.Write the first-order conditions for the optimal stocks of health compactly as

VtGt = t-i (r - rt- +t), t < n, (12)

VnGn = rn,_l(r + 1). (13)

Note that Equation (13) follows from the condition for optimal gross investment inperiod n - 1. An investment in that period yields returns in one period only (period n)since the individual dies after period n. Put differently, the person behaves as if the rateof depreciation on the stock of health is equal to 1 in period n.

10 Readers seeking a definitive and path-breaking treatment of the allocation of goods and time over the lifecycle, should consult Ghez's pioneering monograph on this topic with Becker [Ghez and Becker (1975)].In the late 1970s and 1980s, there was a tremendous growth in the literature on life-cycle labor supply andconsumption demand using the concept of demand functions that hold the marginal utility of wealth constant.All this literature can be traced to Ghez's treatment of the topic.

My publications on the demand for health were outgrowths of my 1970 ColumbiaUniversity Ph.D. dissertation. While I was writing my dissertation, my friend and fellowPh.D. candidate, Gilbert R. Ghez, pointed out that the determination of optimal length oflife could be viewed as an iterative process. I learned a great deal from him, and I oftenspent a long time working through the implications of his comments.l° It has taken mealmost thirty years to work through his comment on the iterative determination of lengthof life. I abandoned this effort many years ago but returned to it when I read Ried's(1996, 1998) reformulation of the selection of the optimal stock of health and lengthof life as a discrete time optimal control problem. Ried (1998, p. 389) writes: "Since[the problem] is a free terminal time problem, one may suspect that a condition for theoptimal length of the planning horizon is missing in the set of necessary conditions ....However, unlike the analogous continuous time problem, the discrete time version failsto provide such an equation. Rather, the optimal final period ... has to be determinedthrough the analysis of a sequence of fixed terminal time problems with the terminaltime varying over a plausible domain." This is the same observation that Ghez made.I offer a proof below. I do not rely on Ried's solution. Instead, I offer a much simplerproof which has a very different implication than the one offered by Ried.

A few preliminaries are in order. First, I assume that the rate of depreciation on thestock of health (,) rises with age. As we shall see in more detail later on, this impliesthat the optimal stock falls with age. Second, I assume that optimal gross investmentin health is positive except in the very last year of life. Third, I define Vt as Wt +(Uht/X)(l + r)t. Hence, Vt is the undiscounted marginal value of the output producedby health capital in period t. Finally, since the output produced by health capital hasa finite upper limit of 8,760 hours in a year, I assume that the marginal product of thestock of health (Gt) diminishes as the stock increases (G,/Hlat < 0).

Consider the maximization problem outlined in Section 2.1 except that the planninghorizon is exogenous. That is, an individual is alive in period n and dead in period n + 1.Write the first-order conditions for the optimal stocks of health compactly as

VtGt = t-i (r - rt- +t), t < n, (12)

VnGn = rn,_l(r + 1). (13)

Note that Equation (13) follows from the condition for optimal gross investment inperiod n - 1. An investment in that period yields returns in one period only (period n)since the individual dies after period n. Put differently, the person behaves as if the rateof depreciation on the stock of health is equal to 1 in period n.

10 Readers seeking a definitive and path-breaking treatment of the allocation of goods and time over the lifecycle, should consult Ghez's pioneering monograph on this topic with Becker [Ghez and Becker (1975)].In the late 1970s and 1980s, there was a tremendous growth in the literature on life-cycle labor supply andconsumption demand using the concept of demand functions that hold the marginal utility of wealth constant.All this literature can be traced to Ghez's treatment of the topic.

I also will make use of the first-order conditions for gross investment in health inperiods 0 and n:

VIGI d2 V2G 2 d VG,(I +r) (1 +r)2 ( +r)n' (14)

In = 0. (15)

In Equation (14), dt is the increase in the stock of health in period t caused by anincrease in gross investment in period 0:

d = 1, dt(t > 1) = (1 -j).j=l

Obviously, gross investment in period n is 0 because the individual will not be alive inperiod n + 1 to collect the returns.

In order for death to take place in period n + 1, H+i < Hmin. Since In = 0,

Hn+t = (1 - ,)Hn. (16)

Hence, for the solution (death after period n) to be fully consistent

Hn+l = (1 - 8n)H, < Hmin. (17)

Suppose that condition (17) is violated. That is, suppose maximization for a fixed num-ber of periods equal to n results in a stock in period n + 1 that exceeds the death stock.Then lifetime utility should be re-maximized under the assumption that the individualwill be alive in period n + 1 but dead in period n + 2. As a first approximation, the setof first-order conditions for Ht (t < n) defined by Equation (12) still must hold so thatthe stock in each of these periods is not affected when the horizon is lengthened by 1period. 11 But the condition for the stock in period n becomes

V,*G* = n, (r - in- + 1), (18)

where asterisks are used because the stock of health in period n when the horizon isn + 1 is not equal to the stock when the horizon is n (see below). Moreover,

Vn*+1Gn +l = 7n(r + 1), (19)

11 This is a first approximation because it assumes that that A does not change when the horizon is extendedby one period. Consider the standard intertemporally separable lifetime utility function in which the currentperiod utility function, lr(ht, Zt), is strictly concave. With full wealth constant, an increase in the horizoncauses to rise. But full wealth increases by Wn+l 2/(l + r) n + l, which causes ). to decline. I assume thatthese two effects exactly offset each other. This assumption is not necessary in the pure investment modeldescribed in Sections 2.5 and 3 because Vt does not depend on X in that model.

358 M. Grossman

I also will make use of the first-order conditions for gross investment in health inperiods 0 and n:

VIGI d2 V2G 2 d VG,(I +r) (1 +r)2 ( +r)n' (14)

In = 0. (15)

In Equation (14), dt is the increase in the stock of health in period t caused by anincrease in gross investment in period 0:

d = 1, dt(t > 1) = (1 -j).j=l

Obviously, gross investment in period n is 0 because the individual will not be alive inperiod n + 1 to collect the returns.

In order for death to take place in period n + 1, H+i < Hmin. Since In = 0,

Hn+t = (1 - ,)Hn. (16)

Hence, for the solution (death after period n) to be fully consistent

Hn+l = (1 - 8n)H, < Hmin. (17)

Suppose that condition (17) is violated. That is, suppose maximization for a fixed num-ber of periods equal to n results in a stock in period n + 1 that exceeds the death stock.Then lifetime utility should be re-maximized under the assumption that the individualwill be alive in period n + 1 but dead in period n + 2. As a first approximation, the setof first-order conditions for Ht (t < n) defined by Equation (12) still must hold so thatthe stock in each of these periods is not affected when the horizon is lengthened by 1period. 11 But the condition for the stock in period n becomes

V,*G* = n, (r - in- + 1), (18)

where asterisks are used because the stock of health in period n when the horizon isn + 1 is not equal to the stock when the horizon is n (see below). Moreover,

Vn*+1Gn +l = 7n(r + 1), (19)

11 This is a first approximation because it assumes that that A does not change when the horizon is extendedby one period. Consider the standard intertemporally separable lifetime utility function in which the currentperiod utility function, lr(ht, Zt), is strictly concave. With full wealth constant, an increase in the horizoncauses to rise. But full wealth increases by Wn+l 2/(l + r) n + l, which causes ). to decline. I assume thatthese two effects exactly offset each other. This assumption is not necessary in the pure investment modeldescribed in Sections 2.5 and 3 because Vt does not depend on X in that model.

358 M. Grossman

In+l = 0, (20)

Hn+2 = (1 - n+l)H+i 1. (21)

If the stock defined by Equation (21) is less than or equal to Hmin, death takes placein period n + 2. If Hn+2 is greater than Hmin, the consumer re-maximizes lifetime util-ity under the assumption that death takes place in period n + 3 (the horizon ends inperiod n +- 2).

I have just described an iterative process for the selection of optimal length of life. Inwords, the process amounts to maximizing lifetime utility for a fixed horizon, checkingto see whether the stock in the period after the horizon ends (the terminal stock) is lessthan or equal to the death stock (Hmin), and adding one period to the horizon and re-maximizing the utility function if the terminal stock exceeds the death stock. 12 I wantto make several comments on this process and its implications. Compare the conditionfor the optimal stock of health in period n when the horizon lasts through period n[Equation (13)] with the condition for the optimal stock in the same period when thehorizon lasts through period n + 1 [Equation (18)]. The supply price of health capitalis smaller in the latter case because 6, < 1.13 Hence, the undiscounted value of themarginal product of health capital in period n when the horizon is n + 1 (V,* G*) must besmaller than the undiscounted value of the marginal product of health capital in periodn when the horizon is n (V,nG). In turn, due to diminishing marginal productivity,the stock of health in period n must rise when the horizon is extended by one period(H* > H). 14

12 If H, 6 Hmin, the utility function is re-maximized after shortening the horizon by 1 period.13 It may appear that the supply price of capital given by the right-hand side of Equation (18) is smaller thanthe one given by the right-hand side of Equation (13) as long as 1 - ,, > -7, 1. But Equation (18) is basedon the assumption 8,,n_ I - 0. The exact form of (18) is

V, G' = r,,l (r + I) - (1 - ,)7r,.

The difference between the right-hand side of this equation and the right-hand side of Equation (13) is -(I -6n)r, < 0.14 While G* is smaller than G,, it is not clear whether V,* is smaller than V,,. The last term can be written

V = W + m,Uh

where U, is the marginal utility of Z,, and m,, is the marginal cost of producing Z,,. The wage rate in periodn (W,,) and m, are not affected when the length of the horizon is increased from n to n + 1. In equilibrium,

Uhn [(r- (l + r))/ G - Wn,

Uh, _ _-_(1 m r) _ _ (l ? _))/G_ - W

u7 m,,

When the individual lives for n + 1 years, the first-order condition for gross invest-ment in period 0 is

V1 G d2 V2G2 d, V* G dn(l-1 * )V *1 G I GA~no~~~ ~ ~ + + + 22 ( -r) (1 + r) 2 ( + + r)" (1 + r. )n+(

Note that the discounted marginal benefits of an investment in period 0 are the samewhether the person dies in period n + 1 or in period n + 2 [compare the right-hand sidesof Equations (14) and (22)] since the marginal cost of an investment in period 0 does notdepend on the length of the horizon. This may seem strange because one term is addedto discounted marginal benefits of an investment in period 0 or in any other period whenthe horizon is extended by one period - the discounted marginal benefit in period n + 1.This term, however, is exactly offset by the reduction in the discounted marginal benefitin period n. The same offset occurs in the discounted marginal benefits of investmentsin every other period except for periods n - 1 and n.

A proof of the last proposition is as follows. The first n - 1 terms on the right-handsides of Equations (14) and (22) are the same. From Equations (13), (18), and (19),

VnGn(r - l,- - ,an)Vn G = (

(1 + r)V+ I G I= V ,,G (1 + n-1)

Hence, the sum of the last two terms on the right-hand side of Equation (22) equals thelast term on the right-hand side of Equation (14): 15

d V,* G (- GnG.

(1 + r)n (1 + r)" +

_ (1 + r)n

Using the last result, one can fully describe the algorithm for the selection of optimallength of life. Maximize the lifetime utility function for a fixed horizon. Check to see

Suppose that

G* G,[7rl (r + 1) -rn(I - 3)c,* = G,,7GL 7r, (r + 1) J

Then Uh / U* = Uhn / Un and V,, = V,. In this case there is no incentive to substitute healthy time for theother commodity in the utility function in period n when the horizon is increased by one period. In other casesthis type of substitution will occur. If it does occur, I assume that the lifetime utility function is separable overtime so that the marginal rate of substitution between h and Z in periods other than period n is not affected.Note the distinction between H + 1 and H,+ . The former stock is the one associated with an n period horizonand I, = 0. The latter stock is the one associated with an n + 1 horizon and In > 0. Clearly, H + l > H,,+ 1.15 In deriving Equation (23) I use the approximation that ,n rn, I 0. If the exact form of Equation (18) isemployed (see footnote 13), the approximation is not necessary.

360 M. Grossmnan

When the individual lives for n + 1 years, the first-order condition for gross invest-ment in period 0 is

V1 G d2 V2G2 d, V* G dn(l-1 * )V *1 G I GA~no~~~ ~ ~ + + + 22 ( -r) (1 + r) 2 ( + + r)" (1 + r. )n+(

Note that the discounted marginal benefits of an investment in period 0 are the samewhether the person dies in period n + 1 or in period n + 2 [compare the right-hand sidesof Equations (14) and (22)] since the marginal cost of an investment in period 0 does notdepend on the length of the horizon. This may seem strange because one term is addedto discounted marginal benefits of an investment in period 0 or in any other period whenthe horizon is extended by one period - the discounted marginal benefit in period n + 1.This term, however, is exactly offset by the reduction in the discounted marginal benefitin period n. The same offset occurs in the discounted marginal benefits of investmentsin every other period except for periods n - 1 and n.

A proof of the last proposition is as follows. The first n - 1 terms on the right-handsides of Equations (14) and (22) are the same. From Equations (13), (18), and (19),

VnGn(r - l,- - ,an)Vn G = (

(1 + r)V+ I G I= V ,,G (1 + n-1)

Hence, the sum of the last two terms on the right-hand side of Equation (22) equals thelast term on the right-hand side of Equation (14): 15

d V,* G (- GnG.

(1 + r)n (1 + r)" +

_ (1 + r)n

Using the last result, one can fully describe the algorithm for the selection of optimallength of life. Maximize the lifetime utility function for a fixed horizon. Check to see

Suppose that

G* G,[7rl (r + 1) -rn(I - 3)c,* = G,,7GL 7r, (r + 1) J

Then Uh / U* = Uhn / Un and V,, = V,. In this case there is no incentive to substitute healthy time for theother commodity in the utility function in period n when the horizon is increased by one period. In other casesthis type of substitution will occur. If it does occur, I assume that the lifetime utility function is separable overtime so that the marginal rate of substitution between h and Z in periods other than period n is not affected.Note the distinction between H + 1 and H,+ . The former stock is the one associated with an n period horizonand I, = 0. The latter stock is the one associated with an n + 1 horizon and In > 0. Clearly, H + l > H,,+ 1.15 In deriving Equation (23) I use the approximation that ,n rn, I 0. If the exact form of Equation (18) isemployed (see footnote 13), the approximation is not necessary.

360 M. Grossmnan

whether the terminal stock is less than or equal to the death stock. If the terminal stockexceeds the death stock, add one period to the horizon and redo the maximization. Theresulting values of the stock of health must be the same in every period except forperiods n and n + 1. The stock of health must be larger in these two periods when thehorizon equals n + 1 than when the horizon equals n. The stock in period t dependson gross investment in period t - 1, with gross investment in previous periods heldconstant. Therefore, gross investment is larger in periods n - 1 and n but the samein every other period when the horizon is increased by one year. A rise in the rate ofdepreciation with age guarantees finite life since for some j 16

Hn+j = (1 - Bn+j-1)Hn+j-l1 Hmin.

I have just addressed a major criticism of my model made by Ehrlich and Chuma(1990). They argue that my analysis does not determine length of life because it". . . does not develop the required terminal (transversality) conditions needed to assurethe consistency of any solutions for the life cycle path of health capital and longevity"[Ehrlich and Chuma (1990, p. 762)]. I have just shown that length of life is determinedas the outcome of an iterative process in which lifetime utility functions with alter-native horizons are maximized. Since the continuous time optimal control techniquesemployed by Ehrlich and Chuma are not my fields of expertise, I invite the reader tostudy their paper and make up his or her own mind on this issue.

As I indicated at the beginning of this subsection, Ried (1996, 1998) offers the samegeneral description of the selection of length of life as an iterative process. He proposes

16 If W,+ 1 = W, and rn = 7rn-1, then Hn+l (the optimal stock when the horizon is n + 1) = H (theoptimal stock when the horizon is n). Hence,

Hn+2 = (1 -n+l)Hn,

H,+{ = (1-8n)Hn.

Since Sn+ > 6n, H,+2 could be smaller than or equal to Hmin, while at the same time Hn could exceedHmin. Note that one addition to the algorithm described is required. Return to the case when maximizationfor a fixed number of periods equal to n results in a stock in period n + 1 that is smaller than or equal to thedeath stock. The consumer should behave as if the rate of depreciation on the stock of health is equal to 1 inperiod n and consult Equation (13) to determine In _ 1 and Hn. Suppose instead that he behaves as if the rateof depreciation is the actual rate in period n (6,, < 1). This is the same rate used by the consumer who dies inperiod n + 2. Denote In*_ as the quantity of gross investment that results from using Equation (18) to selectthe optimal stock in period n. Under the alternative decision rule, the stock in period n + 1 could exceed thedeath stock. The difference in the stock in period n + 1 that results from these two alternative decision rulesis

Hn+- Hn+ = In-I + n-l(In-l

where I assume that In (gross investment in period n when death takes place in period n + 2) equals In_1(gross investment in period n - 1 when death takes place in period n + 1). This difference falls as In*- falls.In turn, In*1 falls as ,, rises with rates of depreciation in all other periods held constant.

whether the terminal stock is less than or equal to the death stock. If the terminal stockexceeds the death stock, add one period to the horizon and redo the maximization. Theresulting values of the stock of health must be the same in every period except forperiods n and n + 1. The stock of health must be larger in these two periods when thehorizon equals n + 1 than when the horizon equals n. The stock in period t dependson gross investment in period t - 1, with gross investment in previous periods heldconstant. Therefore, gross investment is larger in periods n - 1 and n but the samein every other period when the horizon is increased by one year. A rise in the rate ofdepreciation with age guarantees finite life since for some j 16

Hn+j = (1 - Bn+j-1)Hn+j-l1 Hmin.

I have just addressed a major criticism of my model made by Ehrlich and Chuma(1990). They argue that my analysis does not determine length of life because it". . . does not develop the required terminal (transversality) conditions needed to assurethe consistency of any solutions for the life cycle path of health capital and longevity"[Ehrlich and Chuma (1990, p. 762)]. I have just shown that length of life is determinedas the outcome of an iterative process in which lifetime utility functions with alter-native horizons are maximized. Since the continuous time optimal control techniquesemployed by Ehrlich and Chuma are not my fields of expertise, I invite the reader tostudy their paper and make up his or her own mind on this issue.

As I indicated at the beginning of this subsection, Ried (1996, 1998) offers the samegeneral description of the selection of length of life as an iterative process. He proposes

16 If W,+ 1 = W, and rn = 7rn-1, then Hn+l (the optimal stock when the horizon is n + 1) = H (theoptimal stock when the horizon is n). Hence,

Hn+2 = (1 -n+l)Hn,

H,+{ = (1-8n)Hn.

Since Sn+ > 6n, H,+2 could be smaller than or equal to Hmin, while at the same time Hn could exceedHmin. Note that one addition to the algorithm described is required. Return to the case when maximizationfor a fixed number of periods equal to n results in a stock in period n + 1 that is smaller than or equal to thedeath stock. The consumer should behave as if the rate of depreciation on the stock of health is equal to 1 inperiod n and consult Equation (13) to determine In _ 1 and Hn. Suppose instead that he behaves as if the rateof depreciation is the actual rate in period n (6,, < 1). This is the same rate used by the consumer who dies inperiod n + 2. Denote In*_ as the quantity of gross investment that results from using Equation (18) to selectthe optimal stock in period n. Under the alternative decision rule, the stock in period n + 1 could exceed thedeath stock. The difference in the stock in period n + 1 that results from these two alternative decision rulesis

Hn+- Hn+ = In-I + n-l(In-l

where I assume that In (gross investment in period n when death takes place in period n + 2) equals In_1(gross investment in period n - 1 when death takes place in period n + 1). This difference falls as In*- falls.In turn, In*1 falls as ,, rises with rates of depreciation in all other periods held constant.

M. Grossman

a solution using extremely complicated discrete time optimal control techniques. Again,I leave the reader to evaluate Ried's solution. But I do want to challenge his conclusionthat "... sufficiently small perturbations of the trajectories of the exogenous variableswill not alter the length of the individual's planning horizon .... [T]he uniqueness as-sumption [about length of life] ensures that the planning horizon may be treated as fixedin comparative dynamic analysis .... Given a fixed length of the individual's life, it isobvious that the mortality aspect is entirely left out of the picture. Thus, the impactof parametric changes upon individual health is confined to the quality of life whichimplies the analysis to deal [sic] with a pure morbidity effect" (p. 389).

In my view it is somewhat unsatisfactory to begin with a model in which length oflife is endogenous but to end up with a result in which length of life does not dependon any of the exogenous variables in the model. This certainly is not an implicationof my analysis of the determination of optimal length of life. In general, differences insuch exogenous variables as the rate of depreciation, initial assets, and the marginal costof investing in health across consumers of the same age will lead to differences in theoptimal length of life. 17

To be concrete, consider two consumers: a and b. Person a faces a higher rate of de-preciation in each period than person b. The two consumers are the same in all otherrespects. Suppose that it is optimal for person a to live for n years (to die in year n + 1).Ried argues that person b also lives for n years because both he and person a use Equa-tion (13) to determine the optimal stock of health in period n. That equation is indepen-dent of the rate of depreciation in period n. Hence, the stock of health in period n is thesame for each consumer. For person a, we have

H, = ( - a)Hna

where the superscript a denotes values of variables for person a. But for person b,

Hnl= (1 - nb)Ha.

Since 3b < a, there is no guarantee that Hb+ ± Hmin. If Hb > Hmin, person b willn+l n+!be alive in period n + 1. He will then use Equation (18), rather than Equation (13), topick his optimal stock in period n. In this case person b will have a larger optimal stockin period n than person a and will have a longer length of life.

Along the same lines parametric differences in the marginal cost of investment inhealth (differences in the marginal cost across people of the same age), differences ininitial assets, and parametric differences in wage rates cause length of life to vary amongindividuals. In general, any variable that raises the optimal stock of health in each period

17 Technically, I am dealing with parametric differences in exogenous variables (differences in exogenousvariables across consumers) as opposed to evolutionary differences in exogenous variables (differences inexogenous variables across time for the same consumer). This distinction goes back to Ghez and Becker

(1975) and is explored in detail by MaCurdy (1981).

M. Grossman

a solution using extremely complicated discrete time optimal control techniques. Again,I leave the reader to evaluate Ried's solution. But I do want to challenge his conclusionthat "... sufficiently small perturbations of the trajectories of the exogenous variableswill not alter the length of the individual's planning horizon .... [T]he uniqueness as-sumption [about length of life] ensures that the planning horizon may be treated as fixedin comparative dynamic analysis .... Given a fixed length of the individual's life, it isobvious that the mortality aspect is entirely left out of the picture. Thus, the impactof parametric changes upon individual health is confined to the quality of life whichimplies the analysis to deal [sic] with a pure morbidity effect" (p. 389).

In my view it is somewhat unsatisfactory to begin with a model in which length oflife is endogenous but to end up with a result in which length of life does not dependon any of the exogenous variables in the model. This certainly is not an implicationof my analysis of the determination of optimal length of life. In general, differences insuch exogenous variables as the rate of depreciation, initial assets, and the marginal costof investing in health across consumers of the same age will lead to differences in theoptimal length of life. 17

To be concrete, consider two consumers: a and b. Person a faces a higher rate of de-preciation in each period than person b. The two consumers are the same in all otherrespects. Suppose that it is optimal for person a to live for n years (to die in year n + 1).Ried argues that person b also lives for n years because both he and person a use Equa-tion (13) to determine the optimal stock of health in period n. That equation is indepen-dent of the rate of depreciation in period n. Hence, the stock of health in period n is thesame for each consumer. For person a, we have

H, = ( - a)Hna

where the superscript a denotes values of variables for person a. But for person b,

Hnl= (1 - nb)Ha.

Since 3b < a, there is no guarantee that Hb+ ± Hmin. If Hb > Hmin, person b willn+l n+!be alive in period n + 1. He will then use Equation (18), rather than Equation (13), topick his optimal stock in period n. In this case person b will have a larger optimal stockin period n than person a and will have a longer length of life.

Along the same lines parametric differences in the marginal cost of investment inhealth (differences in the marginal cost across people of the same age), differences ininitial assets, and parametric differences in wage rates cause length of life to vary amongindividuals. In general, any variable that raises the optimal stock of health in each period

17 Technically, I am dealing with parametric differences in exogenous variables (differences in exogenousvariables across consumers) as opposed to evolutionary differences in exogenous variables (differences inexogenous variables across time for the same consumer). This distinction goes back to Ghez and Becker

(1975) and is explored in detail by MaCurdy (1981).

of life also tends to prolong length of life. 18 Thus, if health is not an inferior commodity,an increase in initial assets or a reduction in the marginal cost of investing in healthinduces a longer optimal life. Persons with higher wage rates have more wealth; takenby itself, this prolongs life. But the relative price of health (the price of ht relative to theprice of Zt) may rise as the wage rate rises. If this occurs and the resulting substitutioneffect outweighs the wealth effect, length of life may fall.

According to Ried, death occurs if Hn+l < Hmin rather than if Hn+l < Hmin. Thelatter condition is the one that I employ, but that does not seem to account for the dif-ference between my analysis and his analysis. Ried's only justification of his result is inthe context of a dynamic model of labor supply. He assumes that a non-negativity con-straint is binding in some period and concludes that marginal changes in any exogenousvariable will fail to bring about positive supply.

Ried's conclusion does not appear to be correct. To see this in the most simple man-ner, consider a static model of the supply of labor, and suppose that the marginal rateof substitution between leisure time and consumption evaluated at zero hours of workis greater than or equal to the market wage rate at the initial wage. Hence, no hours aresupplied to the market. Now suppose that the wage rate rises. If the marginal rate ofsubstitution at zero hours equaled the old wage, hours of work will rise above zero. Ifthe marginal rate of substitution at zero hours exceeded the old wage, hours could stillrise above zero if the marginal rate of substitution at zero hours is smaller than the newwage. By the same reasoning, while not every parametric reduction in the rate of depre-ciation on the stock of health will increase optimal length of life (see footnote 18), somereductions surely will do so. I stand by my statement that it is somewhat unsatisfactoryto begin with a model in which length of life is endogenous and end up with a result inwhich length of life does not depend on any of the exogenous variables in the model.

2.4. "Bang-bang" equilibrium

Ehrlich and Chuma (1990) assert that my "key assumption that health investment is pro-duced through a constant-returns-to-scale ... technology introduces a type of indetermi-nacy ('bang-bang') problem with respect to optimal investment and health maintenance

18 Consider two people who face the same rate of depreciation in each period. Person b has higher initialassets than person a and picks a larger stock in each period. Suppose that person a dies after period n. Hence,

Ha+l = ( - )Ha

Hi n -

Suppose that person b, like person a, invests nothing in period n. Then

Hb+l= ( -)Hnb

Clearly, Hb+l > Hna+l since Hb

But both people could die in period n + 1 if Hna+l < Hmin This isa necessary, but not a sufficient, condition. For this reason, I use the term "tends" in the text. This ambiguityis removed if the condition for death is defined by Hn+1 = Hmin. That definition is, however, unsatisfactorybecause the rate of depreciation in period n does not guarantee that it is satisfied. That is, death takes place inn + 1 if 8n ) (Hn - Hmin)/Hn.

of life also tends to prolong length of life. 18 Thus, if health is not an inferior commodity,an increase in initial assets or a reduction in the marginal cost of investing in healthinduces a longer optimal life. Persons with higher wage rates have more wealth; takenby itself, this prolongs life. But the relative price of health (the price of ht relative to theprice of Zt) may rise as the wage rate rises. If this occurs and the resulting substitutioneffect outweighs the wealth effect, length of life may fall.

According to Ried, death occurs if Hn+l < Hmin rather than if Hn+l < Hmin. Thelatter condition is the one that I employ, but that does not seem to account for the dif-ference between my analysis and his analysis. Ried's only justification of his result is inthe context of a dynamic model of labor supply. He assumes that a non-negativity con-straint is binding in some period and concludes that marginal changes in any exogenousvariable will fail to bring about positive supply.

Ried's conclusion does not appear to be correct. To see this in the most simple man-ner, consider a static model of the supply of labor, and suppose that the marginal rateof substitution between leisure time and consumption evaluated at zero hours of workis greater than or equal to the market wage rate at the initial wage. Hence, no hours aresupplied to the market. Now suppose that the wage rate rises. If the marginal rate ofsubstitution at zero hours equaled the old wage, hours of work will rise above zero. Ifthe marginal rate of substitution at zero hours exceeded the old wage, hours could stillrise above zero if the marginal rate of substitution at zero hours is smaller than the newwage. By the same reasoning, while not every parametric reduction in the rate of depre-ciation on the stock of health will increase optimal length of life (see footnote 18), somereductions surely will do so. I stand by my statement that it is somewhat unsatisfactoryto begin with a model in which length of life is endogenous and end up with a result inwhich length of life does not depend on any of the exogenous variables in the model.

2.4. "Bang-bang" equilibrium

Ehrlich and Chuma (1990) assert that my "key assumption that health investment is pro-duced through a constant-returns-to-scale ... technology introduces a type of indetermi-nacy ('bang-bang') problem with respect to optimal investment and health maintenance

18 Consider two people who face the same rate of depreciation in each period. Person b has higher initialassets than person a and picks a larger stock in each period. Suppose that person a dies after period n. Hence,

Ha+l = ( - )Ha

Hi n -

Suppose that person b, like person a, invests nothing in period n. Then

Hb+l= ( -)Hnb

Clearly, Hb+l > Hna+l since Hb

But both people could die in period n + 1 if Hna+l < Hmin This isa necessary, but not a sufficient, condition. For this reason, I use the term "tends" in the text. This ambiguityis removed if the condition for death is defined by Hn+1 = Hmin. That definition is, however, unsatisfactorybecause the rate of depreciation in period n does not guarantee that it is satisfied. That is, death takes place inn + 1 if 8n ) (Hn - Hmin)/Hn.

choices. ... [This limitation precludes] a systematic resolution of the choice of both(their italics) optimal health paths and longevity. ... Later contributions to the literaturespawned by Grossman ... suffer in various degrees from these shortcomings. ... Underthe linear production process assumed by Grossman, the marginal cost of investmentwould be constant, and no interior equilibrium for investment would generally exist"(pp. 762, 764, 768). 19

Ried (1998) addresses this criticism by noting that an infinite rate of investment isnot consistent with equilibrium. Because Ehrlich and Chuma's criticism appears to beso damaging and Ried's treatment of it is brief and not convincing, I want to deal with itbefore proceeding to examine responses of health, gross investment, and health inputs toevolutionary (life-cycle) and parametric variations in key exogenous variables. Ehrlichand Chuma's point is as follows. Suppose that the rate of depreciation on the stock ofhealth is equal to zero at every age, suppose that the marginal cost of gross investmentin health does not depend on the amount of investment, and suppose that none of theother exogenous variables in the model is a function of age. 20 Then the stock of healthis constant over time (net investment is zero). Any discrepancy between the initial stockand the optimal stock is erased in the initial period. In a continuous time model, thismeans an infinite rate of investment to close the gap followed by no investment after that.If the rate of depreciation is positive and constant, the discrepancy between the initialand optimal stock is still eliminated in the initial period. After that, gross investment ispositive, constant, and equal to total depreciation; while net investment is zero.

To avoid the "bang-bang" equilibrium (an infinite rate of investment to eliminate thediscrepancy between the initial and the desired stock followed by no investment if therate of depreciation is zero), Ehrlich and Chuma assume that the production function ofgross investment in health exhibits diminishing returns to scale. Thus, the marginal costof gross or net investment is a positive function of the amount of investment. Given this,there is an incentive to reach the desired stock gradually rather than instantaneouslysince the cost of gradual adjustment is smaller than the cost of instantaneous adjust-ment.

The introduction of diminishing returns to scale greatly complicates the model be-cause the marginal cost of gross investment and its percentage rate of change over timebecome endogenous variables that depend on the quantity of investment and its rate ofchange. In Section 6, I show that the structural demand function for the stock of healthat age t in a model with costs of adjustment is one in which Ht depends on the stockat age t + 1 and the stock at age t - 1. The solution of this second-order differenceequation results in a reduced form demand function in which the stock at age t dependson all past and future values of the exogenous variables. This makes theoretical andeconometric analysis very difficult.

19 For an earlier criticism of my model along the same lines, see Usher (1975).20 In addition, I assume that the market rate of interest is equal to the rate of time preference for the present.See Section 4 for a definition of time preference.

364 M. Grossman

choices. ... [This limitation precludes] a systematic resolution of the choice of both(their italics) optimal health paths and longevity. ... Later contributions to the literaturespawned by Grossman ... suffer in various degrees from these shortcomings. ... Underthe linear production process assumed by Grossman, the marginal cost of investmentwould be constant, and no interior equilibrium for investment would generally exist"(pp. 762, 764, 768). 19

Ried (1998) addresses this criticism by noting that an infinite rate of investment isnot consistent with equilibrium. Because Ehrlich and Chuma's criticism appears to beso damaging and Ried's treatment of it is brief and not convincing, I want to deal with itbefore proceeding to examine responses of health, gross investment, and health inputs toevolutionary (life-cycle) and parametric variations in key exogenous variables. Ehrlichand Chuma's point is as follows. Suppose that the rate of depreciation on the stock ofhealth is equal to zero at every age, suppose that the marginal cost of gross investmentin health does not depend on the amount of investment, and suppose that none of theother exogenous variables in the model is a function of age. 20 Then the stock of healthis constant over time (net investment is zero). Any discrepancy between the initial stockand the optimal stock is erased in the initial period. In a continuous time model, thismeans an infinite rate of investment to close the gap followed by no investment after that.If the rate of depreciation is positive and constant, the discrepancy between the initialand optimal stock is still eliminated in the initial period. After that, gross investment ispositive, constant, and equal to total depreciation; while net investment is zero.

To avoid the "bang-bang" equilibrium (an infinite rate of investment to eliminate thediscrepancy between the initial and the desired stock followed by no investment if therate of depreciation is zero), Ehrlich and Chuma assume that the production function ofgross investment in health exhibits diminishing returns to scale. Thus, the marginal costof gross or net investment is a positive function of the amount of investment. Given this,there is an incentive to reach the desired stock gradually rather than instantaneouslysince the cost of gradual adjustment is smaller than the cost of instantaneous adjust-ment.

The introduction of diminishing returns to scale greatly complicates the model be-cause the marginal cost of gross investment and its percentage rate of change over timebecome endogenous variables that depend on the quantity of investment and its rate ofchange. In Section 6, I show that the structural demand function for the stock of healthat age t in a model with costs of adjustment is one in which Ht depends on the stockat age t + 1 and the stock at age t - 1. The solution of this second-order differenceequation results in a reduced form demand function in which the stock at age t dependson all past and future values of the exogenous variables. This makes theoretical andeconometric analysis very difficult.

19 For an earlier criticism of my model along the same lines, see Usher (1975).20 In addition, I assume that the market rate of interest is equal to the rate of time preference for the present.See Section 4 for a definition of time preference.

364 M. Grossman

Are the modifications introduced by Ehrlich and Chuma really necessary? In my viewthe answer is no. The focus of my theoretical and empirical work and that of otherswho have adopted my framework [Cropper (1977), Muurinen (1982), Wagstaff (1986)]certainly is not on discrepancies between the inherited or initial stock and the desiredstock. I am willing to assume that consumers reach their desired stocks instantaneouslyin order to get sharp predictions that are subject to empirical testing. Gross investment ispositive (but net investment is zero) if the rate of depreciation is positive but constant inmy model. In the Ehrlich-Chuma model, net investment can be positive in this situation.In both models consumers choose an infinite life. In both models life is finite and thestock of health varies over the life cycle if the rate of depreciation is a positive functionof age. In my model positive net investment during certain stages of the life cycle is notruled out. For example, the rate of depreciation might be negatively correlated with ageat early stages of the life cycle. The stock of health would be rising and net investmentwould be positive during this stage of the life cycle.

More fundamentally, Ehrlich and Chuma introduce rising marginal cost of investmentto remove an indeterminacy that really does not exist. In Figure 1 of their paper (p. 768),they plot the marginal cost of an investment in health as of age t and the discountedmarginal benefits of this investment as functions of the quantity of investment. Thediscounted marginal benefit function is independent of the rate of investment. Therefore,no interior equilibrium exists for investment unless the marginal cost function slopesupward. This is the basis of their claim that my model does not determine optimalinvestment because marginal cost does not depend on investment.

Why, however, is the discounted marginal benefit function independent of the amountof investment? In a personal communication, Ehrlich informed me that this is becausethe marginal product of the stock of health at age t does not depend on the amountof investment at age t. Surely that is correct. But an increase in It raises the stock ofhealth in all future periods. Since the marginal product of health capital diminishes asthe stock rises, discounted marginal benefits must fall. Hence, the discounted marginalbenefit function slopes downward, and an interior equilibrium for gross investment inperiod t clearly is possible even if the marginal cost of gross investment is constant.

Since discounted marginal benefits are positive when gross investment is zero, thediscounted marginal benefit function intersects the vertical axis.21 Thus corner solutionsfor gross investment are not ruled out in my model. One such solution occurs if the rate

21 This follows because

aht+j _ ah,+j aHt+j

aI, aHt+j alor

aht+jl = Gt+j(1 - St+)( - St+2) . (1 - St+j-)= Gt+jdt+j .

Clearly, dt+j is positive and finite when It equals zero. Moreover Gt+j is positive and finite when It equalszero as long as Ht+j is positive and finite.

Are the modifications introduced by Ehrlich and Chuma really necessary? In my viewthe answer is no. The focus of my theoretical and empirical work and that of otherswho have adopted my framework [Cropper (1977), Muurinen (1982), Wagstaff (1986)]certainly is not on discrepancies between the inherited or initial stock and the desiredstock. I am willing to assume that consumers reach their desired stocks instantaneouslyin order to get sharp predictions that are subject to empirical testing. Gross investment ispositive (but net investment is zero) if the rate of depreciation is positive but constant inmy model. In the Ehrlich-Chuma model, net investment can be positive in this situation.In both models consumers choose an infinite life. In both models life is finite and thestock of health varies over the life cycle if the rate of depreciation is a positive functionof age. In my model positive net investment during certain stages of the life cycle is notruled out. For example, the rate of depreciation might be negatively correlated with ageat early stages of the life cycle. The stock of health would be rising and net investmentwould be positive during this stage of the life cycle.

More fundamentally, Ehrlich and Chuma introduce rising marginal cost of investmentto remove an indeterminacy that really does not exist. In Figure 1 of their paper (p. 768),they plot the marginal cost of an investment in health as of age t and the discountedmarginal benefits of this investment as functions of the quantity of investment. Thediscounted marginal benefit function is independent of the rate of investment. Therefore,no interior equilibrium exists for investment unless the marginal cost function slopesupward. This is the basis of their claim that my model does not determine optimalinvestment because marginal cost does not depend on investment.

Why, however, is the discounted marginal benefit function independent of the amountof investment? In a personal communication, Ehrlich informed me that this is becausethe marginal product of the stock of health at age t does not depend on the amountof investment at age t. Surely that is correct. But an increase in It raises the stock ofhealth in all future periods. Since the marginal product of health capital diminishes asthe stock rises, discounted marginal benefits must fall. Hence, the discounted marginalbenefit function slopes downward, and an interior equilibrium for gross investment inperiod t clearly is possible even if the marginal cost of gross investment is constant.

Since discounted marginal benefits are positive when gross investment is zero, thediscounted marginal benefit function intersects the vertical axis.21 Thus corner solutionsfor gross investment are not ruled out in my model. One such solution occurs if the rate

21 This follows because

aht+j _ ah,+j aHt+j

aI, aHt+j alor

aht+jl = Gt+j(1 - St+)( - St+2) . (1 - St+j-)= Gt+jdt+j .

Clearly, dt+j is positive and finite when It equals zero. Moreover Gt+j is positive and finite when It equalszero as long as Ht+j is positive and finite.

of depreciation on the stock of health equals zero in every period. Given positive ratesof depreciation, corner solutions still are possible in periods other than the last periodof life because the marginal cost of gross investment could exceed discounted marginalbenefits for all positive quantities of investment. I explicitly rule out corner solutionswhen depreciation rates are positive, and Ried and other persons who have used mymodel also rule them out. Corner solutions are possible in the Ehrlich-Chuma model ifthe marginal cost function of gross investment intersects the vertical axis. Ehrlich and

Chuma rule out corner solutions by assuming that the marginal cost function passesthrough the origin.

To summarize, unlike Ried, I conclude that exogenous variations in the marginal

cost and marginal benefit of an investment in health cause optimal length of life tovary. Unlike Ehrlich and Chuma, I conclude that my 1972 model provides a simple butlogically consistent framework for studying optimal health paths and longevity. At thesame time I want to recognize the value of Ried's emphasis on the determination ofoptimal length of life as the outcome of an iterative process in a discrete time model.I also want to recognize the value of Ehrlich and Chuma's model in cases when there aregood reasons to assume that the marginal cost of investment in health is not constant.

2.5. Special cases

Equation (11) determines the optimal stock of health in any period other than the lastperiod of life. A slightly different form of that equation emerges if both sides are dividedby the marginal cost of gross investment:

yt + at = r - t- 1 + t. (24)

Here t =-Wt Gt /rt, defines the marginal monetary return on an investment in healthand at [(Uht /X)(l + r)t Gt]/nr- defines the psychic rate of return. 2 2 In equilibrium,the total rate of return to an investment in health must equal the user cost of capital in

terms of the price of gross investment. The latter variable is defined as the sum of thereal-own rate of interest and the rate of depreciation and may be termed the opportunitycost of health capital.

In Sections 3 and 4, Equation (24) is used to study the responses of the stock ofhealth, gross investment in health, and health inputs to variations in exogenous vari-

ables. Instead of doing this in the context of the general model developed so far, I deal

with two special cases: a pure investment model and a pure consumption model. In theformer model the psychic rate of return is zero, while in the latter the monetary rate ofreturn is zero. There are two reasons for taking this approach. One involves an appeal

22 The corresponding condition for the optimal stock in the last period of life, period n, is

yn + a, = r + 1.

366 M. Grossman

of depreciation on the stock of health equals zero in every period. Given positive ratesof depreciation, corner solutions still are possible in periods other than the last periodof life because the marginal cost of gross investment could exceed discounted marginalbenefits for all positive quantities of investment. I explicitly rule out corner solutionswhen depreciation rates are positive, and Ried and other persons who have used mymodel also rule them out. Corner solutions are possible in the Ehrlich-Chuma model ifthe marginal cost function of gross investment intersects the vertical axis. Ehrlich and

Chuma rule out corner solutions by assuming that the marginal cost function passesthrough the origin.

To summarize, unlike Ried, I conclude that exogenous variations in the marginal

cost and marginal benefit of an investment in health cause optimal length of life tovary. Unlike Ehrlich and Chuma, I conclude that my 1972 model provides a simple butlogically consistent framework for studying optimal health paths and longevity. At thesame time I want to recognize the value of Ried's emphasis on the determination ofoptimal length of life as the outcome of an iterative process in a discrete time model.I also want to recognize the value of Ehrlich and Chuma's model in cases when there aregood reasons to assume that the marginal cost of investment in health is not constant.

2.5. Special cases

Equation (11) determines the optimal stock of health in any period other than the lastperiod of life. A slightly different form of that equation emerges if both sides are dividedby the marginal cost of gross investment:

yt + at = r - t- 1 + t. (24)

Here t =-Wt Gt /rt, defines the marginal monetary return on an investment in healthand at [(Uht /X)(l + r)t Gt]/nr- defines the psychic rate of return. 2 2 In equilibrium,the total rate of return to an investment in health must equal the user cost of capital in

terms of the price of gross investment. The latter variable is defined as the sum of thereal-own rate of interest and the rate of depreciation and may be termed the opportunitycost of health capital.

In Sections 3 and 4, Equation (24) is used to study the responses of the stock ofhealth, gross investment in health, and health inputs to variations in exogenous vari-

ables. Instead of doing this in the context of the general model developed so far, I deal

with two special cases: a pure investment model and a pure consumption model. In theformer model the psychic rate of return is zero, while in the latter the monetary rate ofreturn is zero. There are two reasons for taking this approach. One involves an appeal

22 The corresponding condition for the optimal stock in the last period of life, period n, is

yn + a, = r + 1.

366 M. Grossman

to simplicity. It is difficult to obtain sharp predictions concerning the effects of changesin exogenous variables in a mixed model in which the stock of health yields both in-vestment and consumption benefits. The second is that most treatments of investmentsin knowledge or human capital other than health capital assume that monetary returnsare large relative to psychic returns. Indeed, Lazear (1977) estimates that the psychicreturns from attending school are negative. Clearly, it is unreasonable to assume thathealth is a source of disutility, and most discussions of investments in infant, child, andadolescent health [see Currie (2000)] stress the consumption benefits of these invest-ments. Nevertheless, I stress the pure investment model because it generates powerfulpredictions from simple analyses and because the consumption aspects of the demandfor health can be incorporated into empirical estimation without much loss in generality.

3. Pure investment model

If healthy time did not enter the utility function directly or if the marginal utility ofhealthy time were equal to zero, health would be solely an investment commodity. Theoptimal amount of Ht (t < n) could then be found by equating the marginal monetaryrate of return on an investment in health to the opportunity cost of capital:

Wt Yt = r - t-1 + 8t. (25)7rt

Similarly, the optimal stock of health in the last period of life would be determined by

n - = r + 1. (26)7rn-

Figure 1 illustrates the determination of the optimal stock of health capital at age t.The demand curve MEC shows the relationship between the stock of health and the rateof return on an investment or the marginal efficiency of health capital. The supply curveS shows the relationship between the stock of health and the cost of capital. Since thereal-own rate of interest (r - :7t-1) and the rate of depreciation are independent of thestock, the supply curve is infinitely elastic. Provided the MEC schedule slopes down-ward, the equilibrium stock is given by Ht , where the supply and demand functionsintersect.

The wage rate and the marginal cost of gross investment do not depend on the stockof health. Therefore, the MEC schedule would be negatively inclined if and only ifthe marginal product of health capital (Gt) diminishes as the stock increases. I havealready assumed diminishing marginal productivity in Section 2 and have justified thisassumption because the output produced by health capital has a finite upper limit of8,760 hours in a year. Figure 2 shows a plausible relationship between the stock of healthand the amount of healthy time. This relationship may be termed a production function

to simplicity. It is difficult to obtain sharp predictions concerning the effects of changesin exogenous variables in a mixed model in which the stock of health yields both in-vestment and consumption benefits. The second is that most treatments of investmentsin knowledge or human capital other than health capital assume that monetary returnsare large relative to psychic returns. Indeed, Lazear (1977) estimates that the psychicreturns from attending school are negative. Clearly, it is unreasonable to assume thathealth is a source of disutility, and most discussions of investments in infant, child, andadolescent health [see Currie (2000)] stress the consumption benefits of these invest-ments. Nevertheless, I stress the pure investment model because it generates powerfulpredictions from simple analyses and because the consumption aspects of the demandfor health can be incorporated into empirical estimation without much loss in generality.

3. Pure investment model

If healthy time did not enter the utility function directly or if the marginal utility ofhealthy time were equal to zero, health would be solely an investment commodity. Theoptimal amount of Ht (t < n) could then be found by equating the marginal monetaryrate of return on an investment in health to the opportunity cost of capital:

Wt Yt = r - t-1 + 8t. (25)7rt

Similarly, the optimal stock of health in the last period of life would be determined by

n - = r + 1. (26)7rn-

Figure 1 illustrates the determination of the optimal stock of health capital at age t.The demand curve MEC shows the relationship between the stock of health and the rateof return on an investment or the marginal efficiency of health capital. The supply curveS shows the relationship between the stock of health and the cost of capital. Since thereal-own rate of interest (r - :7t-1) and the rate of depreciation are independent of thestock, the supply curve is infinitely elastic. Provided the MEC schedule slopes down-ward, the equilibrium stock is given by Ht , where the supply and demand functionsintersect.

The wage rate and the marginal cost of gross investment do not depend on the stockof health. Therefore, the MEC schedule would be negatively inclined if and only ifthe marginal product of health capital (Gt) diminishes as the stock increases. I havealready assumed diminishing marginal productivity in Section 2 and have justified thisassumption because the output produced by health capital has a finite upper limit of8,760 hours in a year. Figure 2 shows a plausible relationship between the stock of healthand the amount of healthy time. This relationship may be termed a production function

M. Grossman

Yt,r - t, + 6

r -t + 6

Figure 1.

Hmin Ht

Figure 2.

of healthy time. The slope of the curve in the figure at any point gives the marginalproduct of health capital. The amount of healthy time equals zero at the death stock,Hmin. Beyond that stock, healthy time increases at a decreasing rate and eventuallyapproaches its upper asymptote as the stock becomes large.

Equations (25) and (26) and Figure 1 enable one to study the responses of the stockof health and gross investment to variations in exogenous variables. As indicated in

M. Grossman

Yt,r - t, + 6

r -t + 6

Figure 1.

Hmin Ht

Figure 2.

M. Grossman

Yt,r - t, + 6

r -t + 6

Figure 1.

Hmin Ht

Figure 2.

Section 2, two types of variations are examined: evolutionary (differences across timefor the same consumer) and parametric (differences across consumers of the same age).In particular I consider evolutionary increases in the rate of depreciation on the stock ofhealth with age and parametric variations in the rate of depreciation, the wage rate, andthe stock of knowledge or human capital exclusive of health capital (E).

3.1. Depreciation rate effects

Consider the effect of an increase in the rate of depreciation on the stock of health ()with age. I have already shown in Section 2 that this factor causes the stock of health tofall with age and produces finite life. Graphically, the supply function in Figure 1 shiftsupward over time or with age, and the optimal stock in each period is lower than in theprevious period.

To quantify the magnitude of percentage rate of decrease in the stock of health overthe life cycle, assume that the wage rate and the marginal cost of gross investment inhealth do not depend on age so that art-l = O0. Differentiate Equation (25) with respectto age to obtain23

Ht = -StEtt. (27)

In this equation, the tilde notation denotes a percentage time or age derivative

dHt 1dt -- , etc.,

and the new symbols are st = St/(r + st), the share of the depreciation rate in the costof health capital; and

a ln Ht a ln Ht

a ln(r + t) aln y

the elasticity of the MEC schedule.Provided the rate of depreciation rises over the life cycle, the stock of health falls with

age. The life cycle profile of gross investment does not, however, simply mirror that ofhealth capital. The reason is that a rise in the rate of depreciation not only reduces theamount of health capital demanded by consumers but also reduces the amount of capitalsupplied to them by a given amount of gross investment. If the change in supply exceedsthe change in demand, individuals have an incentive to close the gap by increasing gross

23 As Ghez and Becker (1975) point out, none of the variables in a discrete time model are differentiablefunctions of time. Equation (27) and other equations involving time or age derivatives are approximationsthat hold exactly in a continuous time model.

Section 2, two types of variations are examined: evolutionary (differences across timefor the same consumer) and parametric (differences across consumers of the same age).In particular I consider evolutionary increases in the rate of depreciation on the stock ofhealth with age and parametric variations in the rate of depreciation, the wage rate, andthe stock of knowledge or human capital exclusive of health capital (E).

3.1. Depreciation rate effects

Consider the effect of an increase in the rate of depreciation on the stock of health ()with age. I have already shown in Section 2 that this factor causes the stock of health tofall with age and produces finite life. Graphically, the supply function in Figure 1 shiftsupward over time or with age, and the optimal stock in each period is lower than in theprevious period.

To quantify the magnitude of percentage rate of decrease in the stock of health overthe life cycle, assume that the wage rate and the marginal cost of gross investment inhealth do not depend on age so that art-l = O0. Differentiate Equation (25) with respectto age to obtain23

Ht = -StEtt. (27)

In this equation, the tilde notation denotes a percentage time or age derivative

dHt 1dt -- , etc.,

and the new symbols are st = St/(r + st), the share of the depreciation rate in the costof health capital; and

a ln Ht a ln Ht

a ln(r + t) aln y

the elasticity of the MEC schedule.Provided the rate of depreciation rises over the life cycle, the stock of health falls with

age. The life cycle profile of gross investment does not, however, simply mirror that ofhealth capital. The reason is that a rise in the rate of depreciation not only reduces theamount of health capital demanded by consumers but also reduces the amount of capitalsupplied to them by a given amount of gross investment. If the change in supply exceedsthe change in demand, individuals have an incentive to close the gap by increasing gross

23 As Ghez and Becker (1975) point out, none of the variables in a discrete time model are differentiablefunctions of time. Equation (27) and other equations involving time or age derivatives are approximationsthat hold exactly in a continuous time model.

investment. On the other hand, if the change in demand exceeds the change in supply,gross investment falls over the life cycle.

To begin to see why gross investment does not necessarily fall over the life cycle, firstconsider the behavior of one component of gross investment, total depreciation (D =t Ht), as the rate of depreciation rises over the life cycle. Assume that the percentage

rate of increase in the rate of depreciation with age () and the elasticity of the MECschedule (et) are constant. Then

> StDt = (1-stc) 0 asE>-.

From the last equation, total depreciation increases with age as long as the elasticity ofthe MEC schedule is less than the reciprocal of the share of the depreciation rate in thecost of health capital. A sufficient condition for this to occur is that is smaller thanone.

If Et and t are constant, the percentage change in gross investment with age is givenby

6(1 - Ste)(6t - StS) + StS (28)

(t - StE)

Since health capital cannot be sold, gross investment cannot be negative. Therefore,St -Ht or St > -ste8. Provided gross investment is positive, the term 8t - ste inthe numerator and denominator of Equation (28) must be positive. Thus, a sufficientcondition for gross investment to be positively correlated with the depreciation rate ise < 1/st. Clearly, t is positive if e < 1.

The important conclusion is reached that, if the elasticity of the MEC schedule isless than one, gross investment and the depreciation rate are positively correlated overthe life cycle, while gross investment and the stock of health are negatively correlated.In fact, the relationship between the amount of healthy time and the stock of healthsuggests that is smaller than one. A general equation for the healthy time productionfunction illustrated by Figure 2 is

h, = 8,760 - BH,- C , (29)

where B and C are positive constants. The corresponding MEC schedule is

In y, = In BC - (C + ) In H + In W + Inr. (30)

The elasticity of this schedule is = 1/(1 + C) < 1 since C > 0.Observe that with the depreciation rate held constant, increases in gross investment

increase the stock of health and the amount of healthy time. But the preceding discussionindicates that because the depreciation rate rises with age it is likely that unhealthy (old)

370 M. Grossman

investment. On the other hand, if the change in demand exceeds the change in supply,gross investment falls over the life cycle.

To begin to see why gross investment does not necessarily fall over the life cycle, firstconsider the behavior of one component of gross investment, total depreciation (D =t Ht), as the rate of depreciation rises over the life cycle. Assume that the percentage

rate of increase in the rate of depreciation with age () and the elasticity of the MECschedule (et) are constant. Then

> StDt = (1-stc) 0 asE>-.

From the last equation, total depreciation increases with age as long as the elasticity ofthe MEC schedule is less than the reciprocal of the share of the depreciation rate in thecost of health capital. A sufficient condition for this to occur is that is smaller thanone.

If Et and t are constant, the percentage change in gross investment with age is givenby

6(1 - Ste)(6t - StS) + StS (28)

(t - StE)

Since health capital cannot be sold, gross investment cannot be negative. Therefore,St -Ht or St > -ste8. Provided gross investment is positive, the term 8t - ste inthe numerator and denominator of Equation (28) must be positive. Thus, a sufficientcondition for gross investment to be positively correlated with the depreciation rate ise < 1/st. Clearly, t is positive if e < 1.

The important conclusion is reached that, if the elasticity of the MEC schedule isless than one, gross investment and the depreciation rate are positively correlated overthe life cycle, while gross investment and the stock of health are negatively correlated.In fact, the relationship between the amount of healthy time and the stock of healthsuggests that is smaller than one. A general equation for the healthy time productionfunction illustrated by Figure 2 is

h, = 8,760 - BH,- C , (29)

where B and C are positive constants. The corresponding MEC schedule is

In y, = In BC - (C + ) In H + In W + Inr. (30)

The elasticity of this schedule is = 1/(1 + C) < 1 since C > 0.Observe that with the depreciation rate held constant, increases in gross investment

increase the stock of health and the amount of healthy time. But the preceding discussionindicates that because the depreciation rate rises with age it is likely that unhealthy (old)

370 M. Grossman

people will make larger gross investments in health than healthy (young) people. Thismeans that sick time (TLt) will be positively correlated with the market good or medicalcare input (Mt) and with the own time input (THt) in the gross investment productionfunction over the life cycle.

The framework used to analyze life cycle variations in depreciation rates can easilybe used to examine the impacts of variations in these rates among persons of the sameage. Assume, for example, a uniform percentage shift in t across persons so that thedepreciation rate function can be written as 6t = 60 exp(6t) where 60 differs amongconsumers. It is clear that such a shift has the same kind of effects as an increase in6t with age. That is, persons of a given age who face relatively high depreciation rateswould simultaneously reduce their demand for health but increase their demand forgross investment if e < 1.

3.2. Market and nonmarket efficiency

Persons who face the same cost of health capital would demand the same amount ofhealth only if the determinants of the rate of return on an investment were held constant.Changes in the value of the marginal product of health capital and the marginal cost ofgross investment shift the MEC schedule and, therefore, alter the quantity of healthcapital demanded even if the cost of capital does not change. The consumer's wage rateand his or her stock of knowledge or human capital other than health capital are the twokey shifters of the MEC schedule.2 4

Since the value of the marginal product of health capital equals WG, an increase inthe wage rate (W) raises the monetary equivalent of the marginal product of a givenstock. Put differently, the higher a person's wage rate the greater is the value to him ofan increase in healthy time. A consumer's wage rate measures his market efficiency orthe rate at which he can convert hours of work into money earnings. Hence, the wage ispositively correlated with the benefits of a reduction in lost time from the production ofmoney earnings due to illness. Moreover, a high wage induces an individual to substi-tute market goods for his own time in the production of commodities. This substitutioncontinues until in equilibrium the monetary value of the marginal product of consump-tion time equals the wage rate. Thus, the benefits from a reduction in time lost fromnonmarket production are also positively correlated with the wage.

If an upward shift in the wage rate had no effect on the marginal cost of gross invest-ment, a 1 percent increase in the wage would increase the rate of return (y) associatedwith a fixed stock of capital by 1 percent. In fact this is not the case because own timeis an input in the gross investment production function. If K is the fraction of the total

24 In this section I deal with uniform shifts in variables that influence the rate of return across persons of thesame age. I also assume that these variables do not vary over the life cycle. Finally, I deal with the partialeffect of an increase in the wage rate or an increase in knowledge capital, measured by the number of yearsof formal schooling completed, on the demand for health and health inputs. For a more general treatment, seeGrossman (1972b, pp. 28-30).

people will make larger gross investments in health than healthy (young) people. Thismeans that sick time (TLt) will be positively correlated with the market good or medicalcare input (Mt) and with the own time input (THt) in the gross investment productionfunction over the life cycle.

The framework used to analyze life cycle variations in depreciation rates can easilybe used to examine the impacts of variations in these rates among persons of the sameage. Assume, for example, a uniform percentage shift in t across persons so that thedepreciation rate function can be written as 6t = 60 exp(6t) where 60 differs amongconsumers. It is clear that such a shift has the same kind of effects as an increase in6t with age. That is, persons of a given age who face relatively high depreciation rateswould simultaneously reduce their demand for health but increase their demand forgross investment if e < 1.

3.2. Market and nonmarket efficiency

Persons who face the same cost of health capital would demand the same amount ofhealth only if the determinants of the rate of return on an investment were held constant.Changes in the value of the marginal product of health capital and the marginal cost ofgross investment shift the MEC schedule and, therefore, alter the quantity of healthcapital demanded even if the cost of capital does not change. The consumer's wage rateand his or her stock of knowledge or human capital other than health capital are the twokey shifters of the MEC schedule.2 4

Since the value of the marginal product of health capital equals WG, an increase inthe wage rate (W) raises the monetary equivalent of the marginal product of a givenstock. Put differently, the higher a person's wage rate the greater is the value to him ofan increase in healthy time. A consumer's wage rate measures his market efficiency orthe rate at which he can convert hours of work into money earnings. Hence, the wage ispositively correlated with the benefits of a reduction in lost time from the production ofmoney earnings due to illness. Moreover, a high wage induces an individual to substi-tute market goods for his own time in the production of commodities. This substitutioncontinues until in equilibrium the monetary value of the marginal product of consump-tion time equals the wage rate. Thus, the benefits from a reduction in time lost fromnonmarket production are also positively correlated with the wage.

If an upward shift in the wage rate had no effect on the marginal cost of gross invest-ment, a 1 percent increase in the wage would increase the rate of return (y) associatedwith a fixed stock of capital by 1 percent. In fact this is not the case because own timeis an input in the gross investment production function. If K is the fraction of the total

24 In this section I deal with uniform shifts in variables that influence the rate of return across persons of thesame age. I also assume that these variables do not vary over the life cycle. Finally, I deal with the partialeffect of an increase in the wage rate or an increase in knowledge capital, measured by the number of yearsof formal schooling completed, on the demand for health and health inputs. For a more general treatment, seeGrossman (1972b, pp. 28-30).

M. Grossman

cost of gross investment accounted for by time, a 1 percent rise in W would increasemarginal cost () by K percent. After one nets out the correlation between W and w,the percentage growth in y would equal 1 - K, which exceeds zero as long as grossinvestment is not produced entirely by time. Hence, the quantity of health capital de-manded rises as the wage rate rises as shown in the formula for the wage elasticity ofcapital:

eHW = (1- K)E. (31)

Although the wage rate and the demand for health or gross investment are positivelyrelated, W has no effect on the amount of gross investment supplied by a given inputof medical care. Therefore, the demand for medical care rises with the wage. If medicalcare and own time are employed in fixed proportions in the gross investment productionfunction, the wage elasticity of M equals the wage elasticity of H. On the other hand,given a positive elasticity of substitution in production (p) between M and TH, M in-creases more rapidly than H because consumers have an incentive to substitute medicalcare for their relatively more expensive own time. This substitution is reflected in theformula for the wage elasticity of medical care:

eMWf = Kcrp + (1 - K). (32)

The preceding analysis can be modified to accommodate situations in which themoney price of medical care is zero for all practical purposes because it is fully fi-nanced by health insurance or by the government, and care is rationed by waiting andtravel time. Suppose that q hours are required to obtain one unit of medical care, sothat the price of care is Wq. In addition, suppose that there are three endogenous inputsin the gross investment production function: M, TH, and a market good (X) whoseacquisition does not require time. Interpret K as the share of the cost of gross invest-ment accounted for by M and TH. Then Equation (31) still holds, and an increase in Wcauses H to increase. Equation (32) becomes

eMw = (1 - K)(e - MX), (33)

where orMX is the partial elasticity of substitution in production between M and the thirdinput, X. If these two inputs are net substitutes in production, orMX is positive. Then

eMW < O as e MX.

In this modified model the wage elasticity of medical care could be negative or zero.This case is relevant in interpreting some of the empirical evidence to be discussedlater.

As indicated in Section 2, I follow Michael (1972, 1973) and Michael and Becker(1973) by assuming that an increase in knowledge capital or human capital other than

M. Grossman

cost of gross investment accounted for by time, a 1 percent rise in W would increasemarginal cost () by K percent. After one nets out the correlation between W and w,the percentage growth in y would equal 1 - K, which exceeds zero as long as grossinvestment is not produced entirely by time. Hence, the quantity of health capital de-manded rises as the wage rate rises as shown in the formula for the wage elasticity ofcapital:

eHW = (1- K)E. (31)

Although the wage rate and the demand for health or gross investment are positivelyrelated, W has no effect on the amount of gross investment supplied by a given inputof medical care. Therefore, the demand for medical care rises with the wage. If medicalcare and own time are employed in fixed proportions in the gross investment productionfunction, the wage elasticity of M equals the wage elasticity of H. On the other hand,given a positive elasticity of substitution in production (p) between M and TH, M in-creases more rapidly than H because consumers have an incentive to substitute medicalcare for their relatively more expensive own time. This substitution is reflected in theformula for the wage elasticity of medical care:

eMWf = Kcrp + (1 - K). (32)

The preceding analysis can be modified to accommodate situations in which themoney price of medical care is zero for all practical purposes because it is fully fi-nanced by health insurance or by the government, and care is rationed by waiting andtravel time. Suppose that q hours are required to obtain one unit of medical care, sothat the price of care is Wq. In addition, suppose that there are three endogenous inputsin the gross investment production function: M, TH, and a market good (X) whoseacquisition does not require time. Interpret K as the share of the cost of gross invest-ment accounted for by M and TH. Then Equation (31) still holds, and an increase in Wcauses H to increase. Equation (32) becomes

eMw = (1 - K)(e - MX), (33)

where orMX is the partial elasticity of substitution in production between M and the thirdinput, X. If these two inputs are net substitutes in production, orMX is positive. Then

eMW < O as e MX.

In this modified model the wage elasticity of medical care could be negative or zero.This case is relevant in interpreting some of the empirical evidence to be discussedlater.

As indicated in Section 2, I follow Michael (1972, 1973) and Michael and Becker(1973) by assuming that an increase in knowledge capital or human capital other than

health capital (E) raises the efficiency of the production process in the nonmarket orhousehold sector, just as an increase in technology raises the efficiency of the produc-tion process in the market sector. I focus on education or years of formal schoolingcompleted as the most important determinant of the stock of human capital. The grossinvestment production function and the production function of the commodity Z arelinear homogeneous in their endogenous inputs [see Equations (3) and (4)]. Therefore,an increase in the exogenous or predetermined stock of human capital can raise outputonly if it raises the marginal products of the endogenous inputs.

Suppose that a one unit increase in E raises the marginal products of M and TH in thegross investment production function by the same percentage (PH). This is the Hicks-or factor-neutrality assumption applied to an increase in technology in the nonmarketsector. Given factor-neutrality, there is no incentive to substitute medical care for owntime as the stock of human capital rises.

Because an increase in E raises the marginal products of the health inputs, it reducesthe quantity of these inputs required to produce a given amount of gross investment.Hence, with no change in input prices, the marginal or average cost of gross investmentfalls. In fact, if a circumflex over a variable denotes a percentage change per unit changein E, one easily shows

= -PH. (34)

With the wage rate held constant, an increase in E would raise the marginal efficiencyof a given stock of health. This causes the MEC schedule in Figure 1 to shift upwardand raises the optimal stock of health.

The percentage increase in the amount of health capital demanded for a one unitincrease in E is given by

H = pE. (35)

Since PH indicates the percentage increase in gross investment supplied by a one unitincrease in E, shifts in this variable would not alter the demand for medical care or owntime if PH equaled H. For example, a person with 10 years of formal schooling mightdemand 3 percent more health than a person with 9 years of formal schooling. If themedical care and own time inputs were held constant, the former individual's one extrayear of schooling might supply him with 3 percent more health. Given this condition,both persons would demand the same amounts of M and TH. As this example illustrates,any effect of a change in E on the demand for medical care or time reflects a positiveor negative differential between H and PH:

M= TH= pH(e- 1). (36)

Equation (36) suggests that the more educated would demand more health but less med-ical care if the elasticity of the MEC schedule were less than one. These patterns are

health capital (E) raises the efficiency of the production process in the nonmarket orhousehold sector, just as an increase in technology raises the efficiency of the produc-tion process in the market sector. I focus on education or years of formal schoolingcompleted as the most important determinant of the stock of human capital. The grossinvestment production function and the production function of the commodity Z arelinear homogeneous in their endogenous inputs [see Equations (3) and (4)]. Therefore,an increase in the exogenous or predetermined stock of human capital can raise outputonly if it raises the marginal products of the endogenous inputs.

Suppose that a one unit increase in E raises the marginal products of M and TH in thegross investment production function by the same percentage (PH). This is the Hicks-or factor-neutrality assumption applied to an increase in technology in the nonmarketsector. Given factor-neutrality, there is no incentive to substitute medical care for owntime as the stock of human capital rises.

Because an increase in E raises the marginal products of the health inputs, it reducesthe quantity of these inputs required to produce a given amount of gross investment.Hence, with no change in input prices, the marginal or average cost of gross investmentfalls. In fact, if a circumflex over a variable denotes a percentage change per unit changein E, one easily shows

= -PH. (34)

With the wage rate held constant, an increase in E would raise the marginal efficiencyof a given stock of health. This causes the MEC schedule in Figure 1 to shift upwardand raises the optimal stock of health.

The percentage increase in the amount of health capital demanded for a one unitincrease in E is given by

H = pE. (35)

Since PH indicates the percentage increase in gross investment supplied by a one unitincrease in E, shifts in this variable would not alter the demand for medical care or owntime if PH equaled H. For example, a person with 10 years of formal schooling mightdemand 3 percent more health than a person with 9 years of formal schooling. If themedical care and own time inputs were held constant, the former individual's one extrayear of schooling might supply him with 3 percent more health. Given this condition,both persons would demand the same amounts of M and TH. As this example illustrates,any effect of a change in E on the demand for medical care or time reflects a positiveor negative differential between H and PH:

M= TH= pH(e- 1). (36)

Equation (36) suggests that the more educated would demand more health but less med-ical care if the elasticity of the MEC schedule were less than one. These patterns are

opposite to those that would be expected in comparing the health and medical care uti-lization of older and younger consumers.

4. Pure consumption model

If the cost of health capital were large relative to the monetary rate of return on an invest-ment in health and if art_ = 0, all t, then Equation (11) or (24) could be approximatedby

UhtGt UHt _ (r + t)A A (1+ r) t

Equation (37) indicates that the monetary equivalent of the marginal utility of healthcapital must equal the discounted user cost of Ht .25 It can be used to highlight thedifferences between the age, wage, or schooling effect in a pure consumption modeland the corresponding effect in a pure investment model. In the following analysis,I assume that the marginal rate of substitution between Ht and Ht+l depends only onHt and Ht+l and that the marginal rate of substitution between Ht and Zt depends onlyon Ht and Zt. I also assume that one plus the market rate of interest is equal to oneplus rate of time preference for the present (the ratio of the marginal utility of Ht to themarginal utility of Ht+l when these two stocks are equal minus one). Some of theseassumptions are relaxed, and a more detailed analysis is presented in Grossman (1972b,Chapter III).

With regard to age-related depreciation rate effects, the elasticity of substitution inconsumption between Ht and Ht+ replaces the elasticity of the MEC schedule in Equa-tions (27) and (28). The quantity of health capital demanded still falls over the life cyclein response to an increase in the rate of depreciation. Gross investment and health in-puts rise with age if the elasticity of substitution between present and future health isless than one.

Since health enters the utility function, health is positively related to wealth in theconsumption model provided it is a superior good. That is, an increase in wealth with nochange in the wage rate or the marginal cost of gross investment causes the quantity ofhealth capital demanded to rise. This effect is absent from the investment model because

25 Solving Equation (37) for the monetary equivalent of the marginal utility of healthy time, one obtains

Uht 7(r +St)/Gt

A (I +r) t

Given diminishing marginal productivity of health capital, the undiscounted price of a healthy hour, r(r +6t)/Gt, would be positively correlated with H or h even if r were constant. Therefore, the consumptiondemand curve would be influenced by scale effects. To emphasize the main issues at stake in the consumptionmodel, I ignore these effects essentially by assuming that Gt is constant. The analysis would not be greatlyaltered if they were introduced.

374 M. Grossman

opposite to those that would be expected in comparing the health and medical care uti-lization of older and younger consumers.

4. Pure consumption model

If the cost of health capital were large relative to the monetary rate of return on an invest-ment in health and if art_ = 0, all t, then Equation (11) or (24) could be approximatedby

UhtGt UHt _ (r + t)A A (1+ r) t

Equation (37) indicates that the monetary equivalent of the marginal utility of healthcapital must equal the discounted user cost of Ht .25 It can be used to highlight thedifferences between the age, wage, or schooling effect in a pure consumption modeland the corresponding effect in a pure investment model. In the following analysis,I assume that the marginal rate of substitution between Ht and Ht+l depends only onHt and Ht+l and that the marginal rate of substitution between Ht and Zt depends onlyon Ht and Zt. I also assume that one plus the market rate of interest is equal to oneplus rate of time preference for the present (the ratio of the marginal utility of Ht to themarginal utility of Ht+l when these two stocks are equal minus one). Some of theseassumptions are relaxed, and a more detailed analysis is presented in Grossman (1972b,Chapter III).

With regard to age-related depreciation rate effects, the elasticity of substitution inconsumption between Ht and Ht+ replaces the elasticity of the MEC schedule in Equa-tions (27) and (28). The quantity of health capital demanded still falls over the life cyclein response to an increase in the rate of depreciation. Gross investment and health in-puts rise with age if the elasticity of substitution between present and future health isless than one.

Since health enters the utility function, health is positively related to wealth in theconsumption model provided it is a superior good. That is, an increase in wealth with nochange in the wage rate or the marginal cost of gross investment causes the quantity ofhealth capital demanded to rise. This effect is absent from the investment model because

25 Solving Equation (37) for the monetary equivalent of the marginal utility of healthy time, one obtains

Uht 7(r +St)/Gt

A (I +r) t

Given diminishing marginal productivity of health capital, the undiscounted price of a healthy hour, r(r +6t)/Gt, would be positively correlated with H or h even if r were constant. Therefore, the consumptiondemand curve would be influenced by scale effects. To emphasize the main issues at stake in the consumptionmodel, I ignore these effects essentially by assuming that Gt is constant. The analysis would not be greatlyaltered if they were introduced.

374 M. Grossman

the marginal efficiency of health capital and the market rate of interest do not dependon wealth.26 Parametric wage variations across persons of the same age induce wealtheffects on the demand for health. Suppose that we abstract from these effects by holdingthe level of utility or real wealth constant. Then the wage elasticity of health is given by

eHW = -(1 - O)(K KZ)orHZ, (38)

where 0 is the share of health in wealth, Kz is the share of total cost of Z accountedfor by time, and cXHZ is the positive elasticity of substitution in consumption between Hand Z.27 Hence,

eHW O as K Kz.

The sign of the wage elasticity is ambiguous because an increase in the wage rateraises the marginal cost of gross investment in health and the marginal cost of Z. If timecosts were relatively more important in the production of health than in the productionof Z, the relative price of health would rise with the wage rate, which would reduce thequantity of health demanded. The reverse would occur if Z were more time intensivethan health. The ambiguity of the wage effect here is in sharp contrast to the situation inthe investment model. In that model, the wage rate would be positively correlated withhealth as long as K were less than one.

Instead of examining a wage effect that holds utility constant, Wagstaff (1986) andZweifel and Breyer (1997) examine a wage effect that holds the marginal utility ofwealth constant. This analysis is feasible only if the current period utility function,\ (H, Z), is strictly concave:

IsHH < O, XP'ZZ <0, 'IHHXZZ -, IHHZ > 0.

With the marginal utility of wealth () held constant, the actual change in health causedby a one percent increase in the wage rate is given by

aH K\IHJZZ - KZ Z\3HZ

a In W - IHHWZZ HZ

26 For a different conclusion, see Ehrlich and Chuma (1990). They argue that wealth effects are present in theinvestment model given rising marginal cost of investment in health and endogenous length of life.27 I assume that the elasticity of substitution in consumption between Ht and Zt is the same in every periodand that the elasticity of substitution between Ht and Zj (t j) does not depend on t or j. The correspondingequation for the wage elasticity of medical care is

eMW = Kap - (1 - 0)(K KZ)rYHZ.

the marginal efficiency of health capital and the market rate of interest do not dependon wealth.26 Parametric wage variations across persons of the same age induce wealtheffects on the demand for health. Suppose that we abstract from these effects by holdingthe level of utility or real wealth constant. Then the wage elasticity of health is given by

eHW = -(1 - O)(K KZ)orHZ, (38)

where 0 is the share of health in wealth, Kz is the share of total cost of Z accountedfor by time, and cXHZ is the positive elasticity of substitution in consumption between Hand Z.27 Hence,

eHW O as K Kz.

The sign of the wage elasticity is ambiguous because an increase in the wage rateraises the marginal cost of gross investment in health and the marginal cost of Z. If timecosts were relatively more important in the production of health than in the productionof Z, the relative price of health would rise with the wage rate, which would reduce thequantity of health demanded. The reverse would occur if Z were more time intensivethan health. The ambiguity of the wage effect here is in sharp contrast to the situation inthe investment model. In that model, the wage rate would be positively correlated withhealth as long as K were less than one.

Instead of examining a wage effect that holds utility constant, Wagstaff (1986) andZweifel and Breyer (1997) examine a wage effect that holds the marginal utility ofwealth constant. This analysis is feasible only if the current period utility function,\ (H, Z), is strictly concave:

IsHH < O, XP'ZZ <0, 'IHHXZZ -, IHHZ > 0.

With the marginal utility of wealth () held constant, the actual change in health causedby a one percent increase in the wage rate is given by

aH K\IHJZZ - KZ Z\3HZ

a In W - IHHWZZ HZ

26 For a different conclusion, see Ehrlich and Chuma (1990). They argue that wealth effects are present in theinvestment model given rising marginal cost of investment in health and endogenous length of life.27 I assume that the elasticity of substitution in consumption between Ht and Zt is the same in every periodand that the elasticity of substitution between Ht and Zj (t j) does not depend on t or j. The correspondingequation for the wage elasticity of medical care is

eMW = Kap - (1 - 0)(K KZ)rYHZ.

Equation (39) is negative if 4 'HZ ) O. The sign of the wage effect is, however, ambigu-ous if gtHZ < 0.28

The human capital parameter in the consumption demand function for health is

H = PrlH + (PH - Pz)(1 - O)aHZ, (40)

where pZ is the percentage increase in the marginal product of the Z commodity'sgoods or time input caused by a one unit increase in E (the negative of the percentagereduction in the marginal or average cost of Z), qrH is the wealth elasticity of demand forhealth, and p = PH + (1 - O)pz is the percentage increase in real wealth as E rises withmoney full wealth and the wage rate held constant. The first term on the right-hand sideof Equation (40) reflects the wealth effect and the second term reflects the substitutioneffect. If E's productivity effect in the gross investment production function is the sameas in the Z production function, then pH = Pz and H reflects the wealth effect alone.In this case, a shift in human capital, measured by years of formal schooling completedor education is "commodity-neutral," to use the term coined by Michael (1972, 1973).If PH > PZ, E is "biased" toward health, its relative price falls, and the wealth andsubstitution effects both operate in the same direction. Consequently, an increase in Edefinitely increases the demand for health. If PH < pZ, E is biased away from health, itsrelative price rises, and the wealth and substitution effects operate in opposite directions.

The human capital parameter in the consumption demand curve for medical care is

M = P(rH - 1) + (PH - pz)[(l - O0)rHZ - 1]. (41)

If shifts in E are commodity-neutral, medical care and education are negatively corre-lated unless NH > 1. If on the other hand, there is a bias in favor of health, these twovariables will still tend to be negatively correlated unless the wealth and price elasticitiesboth exceed one.2 9

The preceding discussion reveals that the analysis of variations in nonmarket produc-tivity in the consumption model differs in two important respects from the correspond-ing analysis in the investment model. In the first place, wealth effects are not relevantin the investment model, as has already been indicated. Of course, health would have apositive wealth elasticity in the investment model if wealthier people faced lower ratesof interest. But the analysis of shifts in education assumes money wealth is fixed. Thus,one could not rationalize the positive relationship between education and health in termsof an association between wealth and the interest rate.

28 I assume that q'Z 'HZ > 1H4 `ZZ so that the pure wealth effect is positive and a reduction in raises

health. When qJHZ is negative, this condition does not guarantee that Equation (39) is negative because Kzcould exceed K.29 The term (1 - O)oHz in Equation (40) or Equation (41) is the compensated or utility-constant price elas-ticity of demand for health.

376 M. Grossman

Equation (39) is negative if 4 'HZ ) O. The sign of the wage effect is, however, ambigu-ous if gtHZ < 0.28

The human capital parameter in the consumption demand function for health is

H = PrlH + (PH - Pz)(1 - O)aHZ, (40)

where pZ is the percentage increase in the marginal product of the Z commodity'sgoods or time input caused by a one unit increase in E (the negative of the percentagereduction in the marginal or average cost of Z), qrH is the wealth elasticity of demand forhealth, and p = PH + (1 - O)pz is the percentage increase in real wealth as E rises withmoney full wealth and the wage rate held constant. The first term on the right-hand sideof Equation (40) reflects the wealth effect and the second term reflects the substitutioneffect. If E's productivity effect in the gross investment production function is the sameas in the Z production function, then pH = Pz and H reflects the wealth effect alone.In this case, a shift in human capital, measured by years of formal schooling completedor education is "commodity-neutral," to use the term coined by Michael (1972, 1973).If PH > PZ, E is "biased" toward health, its relative price falls, and the wealth andsubstitution effects both operate in the same direction. Consequently, an increase in Edefinitely increases the demand for health. If PH < pZ, E is biased away from health, itsrelative price rises, and the wealth and substitution effects operate in opposite directions.

The human capital parameter in the consumption demand curve for medical care is

M = P(rH - 1) + (PH - pz)[(l - O0)rHZ - 1]. (41)

If shifts in E are commodity-neutral, medical care and education are negatively corre-lated unless NH > 1. If on the other hand, there is a bias in favor of health, these twovariables will still tend to be negatively correlated unless the wealth and price elasticitiesboth exceed one.2 9

The preceding discussion reveals that the analysis of variations in nonmarket produc-tivity in the consumption model differs in two important respects from the correspond-ing analysis in the investment model. In the first place, wealth effects are not relevantin the investment model, as has already been indicated. Of course, health would have apositive wealth elasticity in the investment model if wealthier people faced lower ratesof interest. But the analysis of shifts in education assumes money wealth is fixed. Thus,one could not rationalize the positive relationship between education and health in termsof an association between wealth and the interest rate.

28 I assume that q'Z 'HZ > 1H4 `ZZ so that the pure wealth effect is positive and a reduction in raises

health. When qJHZ is negative, this condition does not guarantee that Equation (39) is negative because Kzcould exceed K.29 The term (1 - O)oHz in Equation (40) or Equation (41) is the compensated or utility-constant price elas-ticity of demand for health.

376 M. Grossman

In the second place, if the investment framework were utilized, then whether or not ashift in human capital is commodity-neutral would be irrelevant in assessing its impacton the demand for health. As long as the rate of interest were independent of education,H and E would be positively correlated.3 0 Put differently, if individuals could alwaysreceive, say, a 5 percent real rate of return on savings deposited in a savings account,then a shift in education would create a gap between the cost of capital and the marginalefficiency of a given stock.

Muurinen (1982) and Van Doorslaer (1987) assume that an increase in education low-ers the rate of depreciation on the stock of health rather than raising productivity in thegross investment production function. This is a less general assumption than the one thatI have made since it rules out schooling effects in the production of nondurable house-hold commodities. In the pure investment model, predictions are very similar whetherschooling raises productivity or lowers the rate of depreciation. 3 1 In the pure consump-tion model the assumption made by Muurinen and Van Doorslaer is difficult to distin-guish from the alternative assumption that pz is zero. Interactions between schoolingand the lagged stock of health in the demand function for current health arise given costsof adjustment in the Muurinen-Van Doorslaer model. These are discussed in Section 6.

5. Empirical testing

In Grossman (1972b, Chapter IV), I present an empirical formulation of the pure in-vestment model, including a detailed outline of the structure and reduced form of thatmodel. I stress the estimation of the investment model rather than the consumptionmodel because the former model generates powerful predictions from simple analysisand more innocuous assumptions. For example, if one uses the investment model, heor she does not have to know whether health is relatively time-intensive to predict theeffect of an increase in the wage rate on the demand for health. Also, he or she doesnot have to know whether education is commodity-neutral to assess the sign of the cor-relation between health and schooling. Moreover, the responsiveness of the quantity ofhealth demanded to changes in its shadow price and the behavior of gross investmentdepend essentially on a single parameter - the elasticity of the MEC schedule. In theconsumption model, on the other hand, three parameters are relevant - the elasticity ofsubstitution in consumption between present and future health, the wealth elasticity ofdemand for health, and the elasticity of substitution in consumption between health andthe Z commodity. Finally, while good health may be a source of utility, it clearly is a

30 relax the assumption that all persons face the same market rate of interest in Section 7.31 If an increase in schooling lowers the rate of depreciation at every age by the same percentage, it isequivalent to a uniform percentage shift in this rate considered, a matter briefly in Section 3.3 and in moredetail in Grossman (1972b, pp. 19, 90). The only difference between this model and the productivity modelis that a value of the elasticity of the MEC schedule smaller than one is a sufficient, but not a necessary,condition for medical care to fall as schooling rises.

In the second place, if the investment framework were utilized, then whether or not ashift in human capital is commodity-neutral would be irrelevant in assessing its impacton the demand for health. As long as the rate of interest were independent of education,H and E would be positively correlated.3 0 Put differently, if individuals could alwaysreceive, say, a 5 percent real rate of return on savings deposited in a savings account,then a shift in education would create a gap between the cost of capital and the marginalefficiency of a given stock.

Muurinen (1982) and Van Doorslaer (1987) assume that an increase in education low-ers the rate of depreciation on the stock of health rather than raising productivity in thegross investment production function. This is a less general assumption than the one thatI have made since it rules out schooling effects in the production of nondurable house-hold commodities. In the pure investment model, predictions are very similar whetherschooling raises productivity or lowers the rate of depreciation. 3 1 In the pure consump-tion model the assumption made by Muurinen and Van Doorslaer is difficult to distin-guish from the alternative assumption that pz is zero. Interactions between schoolingand the lagged stock of health in the demand function for current health arise given costsof adjustment in the Muurinen-Van Doorslaer model. These are discussed in Section 6.

5. Empirical testing

In Grossman (1972b, Chapter IV), I present an empirical formulation of the pure in-vestment model, including a detailed outline of the structure and reduced form of thatmodel. I stress the estimation of the investment model rather than the consumptionmodel because the former model generates powerful predictions from simple analysisand more innocuous assumptions. For example, if one uses the investment model, heor she does not have to know whether health is relatively time-intensive to predict theeffect of an increase in the wage rate on the demand for health. Also, he or she doesnot have to know whether education is commodity-neutral to assess the sign of the cor-relation between health and schooling. Moreover, the responsiveness of the quantity ofhealth demanded to changes in its shadow price and the behavior of gross investmentdepend essentially on a single parameter - the elasticity of the MEC schedule. In theconsumption model, on the other hand, three parameters are relevant - the elasticity ofsubstitution in consumption between present and future health, the wealth elasticity ofdemand for health, and the elasticity of substitution in consumption between health andthe Z commodity. Finally, while good health may be a source of utility, it clearly is a

30 relax the assumption that all persons face the same market rate of interest in Section 7.31 If an increase in schooling lowers the rate of depreciation at every age by the same percentage, it isequivalent to a uniform percentage shift in this rate considered, a matter briefly in Section 3.3 and in moredetail in Grossman (1972b, pp. 19, 90). The only difference between this model and the productivity modelis that a value of the elasticity of the MEC schedule smaller than one is a sufficient, but not a necessary,condition for medical care to fall as schooling rises.

M. Grossman

source of earnings. The following formulation is oriented toward the investment model,yet I also offer two tests to distinguish the investment model from the consumptionmodel.

5.1. Structure and reducedform

With the production function of healthy time given by Equation (29), make use ofthree basic structural equations (intercepts are suppressed):

In H = In Wt - e In rt - e In St, (42)

lnst = InSo + t, (43)

ln t _ lnHt + n(l + H,/t) = pHE + (1 - K) lnM, + K lnTHt. (44)

Equation (42) is the demand function for the stock of health and is obtained by solvingEquation (30) for In H. The equation contains the assumption that the real-own rateof interest is equal to zero. Equation (43) is the depreciation rate function. Equation(44) contains the identity that gross investment equals net investment plus depreciationand assumes that the gross investment production function is a member of the Cobb-Douglas class.

These three equations and the least-cost equilibrium condition that the ratio of themarginal product of medical care to the marginal product of time must equal the ratio ofthe price of medical care to the wage rate generate the following reduced form demandcurves for health and medical care:

In Ht = (1 - K)E In Wt - (1 - K)E In P + pHE - Et - In So, (45)

In Mt = [(1 - K)E + K] In Wt - [(1 - K) + K] In Pt + PH ( - 1)E

+ (1 - )t + (1 - ) In o + ln(l + Ht/t). (46)

If the absolute value of the rate of net disinvestment (Ht) were small relative to the rateof depreciation, the last term on the right-hand side of Equation (46) could be ignored. 3 2

Then Equations (45) and (46) would express the two main endogenous variables in thesystem as functions of four variables that are treated as exogenous within the context ofthis model - the wage rate, the price of medical care, the stock of human capital, and

32 Recall that 8t > -Ht since gross investment must be positive. Given that the real rate of interest is zero

_Ht z dt8f ' dt

Since E is likely to be smaller than one and the square of the rate of depreciation is small for modest rates ofdepreciation, St is likely to be much larger than -H.

M. Grossman

source of earnings. The following formulation is oriented toward the investment model,yet I also offer two tests to distinguish the investment model from the consumptionmodel.

5.1. Structure and reducedform

With the production function of healthy time given by Equation (29), make use ofthree basic structural equations (intercepts are suppressed):

In H = In Wt - e In rt - e In St, (42)

lnst = InSo + t, (43)

ln t _ lnHt + n(l + H,/t) = pHE + (1 - K) lnM, + K lnTHt. (44)

Equation (42) is the demand function for the stock of health and is obtained by solvingEquation (30) for In H. The equation contains the assumption that the real-own rateof interest is equal to zero. Equation (43) is the depreciation rate function. Equation(44) contains the identity that gross investment equals net investment plus depreciationand assumes that the gross investment production function is a member of the Cobb-Douglas class.

These three equations and the least-cost equilibrium condition that the ratio of themarginal product of medical care to the marginal product of time must equal the ratio ofthe price of medical care to the wage rate generate the following reduced form demandcurves for health and medical care:

In Ht = (1 - K)E In Wt - (1 - K)E In P + pHE - Et - In So, (45)

In Mt = [(1 - K)E + K] In Wt - [(1 - K) + K] In Pt + PH ( - 1)E

+ (1 - )t + (1 - ) In o + ln(l + Ht/t). (46)

If the absolute value of the rate of net disinvestment (Ht) were small relative to the rateof depreciation, the last term on the right-hand side of Equation (46) could be ignored. 3 2

Then Equations (45) and (46) would express the two main endogenous variables in thesystem as functions of four variables that are treated as exogenous within the context ofthis model - the wage rate, the price of medical care, the stock of human capital, and

32 Recall that 8t > -Ht since gross investment must be positive. Given that the real rate of interest is zero

_Ht z dt8f ' dt

Since E is likely to be smaller than one and the square of the rate of depreciation is small for modest rates ofdepreciation, St is likely to be much larger than -H.

Ch. 7 The Human Capital Model

age - and one variable that is unobserved - the rate of depreciation in the initial period.With age subscripts suppressed the estimating equations become

InH = BInW+BplnP+BEE+Btt+ul, (47)

InM = BM In W + BpMln P + BEME+BtMt+u2, (48)

where Bw = (1 - K)e, et cetera, ul = - In 0 and u2 = (1 - )In o. The investmentmodel predicts Bw > 0, Bp < 0, BE > 0, Bt < O, BWM > 0, and BPM < 0. In addi-tion, if s < 1, BEM < 0 and BtM > 0.

The variables u and u2 represent disturbance terms in the reduced form equations.These terms are present because depreciation rates vary among people of the same age,and such variations cannot be measured empirically. Provided In 80 were not correlatedwith the independent variables in (47) and (48), u l and u2 would not be correlated withthese variables. Therefore, the equations could be estimated by ordinary least squares.

The assumption that the real-own rate of interest equals zero can be justified by notingthat wage rates rise with age, at least during most stages of the life cycle. If the wageis growing at a constant percentage rate of W, then rt = KW, all t. So the assumptionimplies r = KW. By eliminating the real rate of interest and postulating that -Ht issmall relative to St, In H and In M are made linear functions of age. If these assumptionsare dropped, the age effect becomes nonlinear.

Since the gross investment production function is a member of the Cobb-Douglasclass, the elasticity of substitution in production between medical care and own time(up) is equal to one, and the share of medical care in the total cost of gross investmentor the elasticity of gross investment with respect to medical care, (1 - K), is constant.If up were not equal to one, the term K in the wage and price elasticities of demand formedical care would be multiplied by this value rather than by one. The wage and priceparameters would not be constant if up were constant but not equal to one, because Kwould depend on W and P. The linear age, price, and wage effects in Equations (47)and (48) are first-order approximations to the true effects.

I have indicated that years of formal schooling completed is the most important deter-minant of the stock of human capital and employ schooling as a proxy for this stock inthe empirical analysis described in Section 5.2. In reality the amount of human capitalacquired by attending school also depends on such variables as the mental ability of thestudent and the quality of the school that he or she attends. If these omitted variables arepositively correlated with schooling and uncorrelated with the other regressors in thedemand function for health, the schooling coefficient is biased upwards. These biasesare more difficult to sign if, for example, mental ability and school quality are correlatedwith the wage rate.3 3

There are two empirical procedures for assessing whether the investment model givesa more adequate representation of people's behavior than the consumption model. In

33 See Section 7 for a detailed analysis of biases in the regression coefficient of schooling due to the omissionof other variables.

Ch. 7 The Human Capital Model

age - and one variable that is unobserved - the rate of depreciation in the initial period.With age subscripts suppressed the estimating equations become

InH = BInW+BplnP+BEE+Btt+ul, (47)

InM = BM In W + BpMln P + BEME+BtMt+u2, (48)

where Bw = (1 - K)e, et cetera, ul = - In 0 and u2 = (1 - )In o. The investmentmodel predicts Bw > 0, Bp < 0, BE > 0, Bt < O, BWM > 0, and BPM < 0. In addi-tion, if s < 1, BEM < 0 and BtM > 0.

The variables u and u2 represent disturbance terms in the reduced form equations.These terms are present because depreciation rates vary among people of the same age,and such variations cannot be measured empirically. Provided In 80 were not correlatedwith the independent variables in (47) and (48), u l and u2 would not be correlated withthese variables. Therefore, the equations could be estimated by ordinary least squares.

The assumption that the real-own rate of interest equals zero can be justified by notingthat wage rates rise with age, at least during most stages of the life cycle. If the wageis growing at a constant percentage rate of W, then rt = KW, all t. So the assumptionimplies r = KW. By eliminating the real rate of interest and postulating that -Ht issmall relative to St, In H and In M are made linear functions of age. If these assumptionsare dropped, the age effect becomes nonlinear.

Since the gross investment production function is a member of the Cobb-Douglasclass, the elasticity of substitution in production between medical care and own time(up) is equal to one, and the share of medical care in the total cost of gross investmentor the elasticity of gross investment with respect to medical care, (1 - K), is constant.If up were not equal to one, the term K in the wage and price elasticities of demand formedical care would be multiplied by this value rather than by one. The wage and priceparameters would not be constant if up were constant but not equal to one, because Kwould depend on W and P. The linear age, price, and wage effects in Equations (47)and (48) are first-order approximations to the true effects.

I have indicated that years of formal schooling completed is the most important deter-minant of the stock of human capital and employ schooling as a proxy for this stock inthe empirical analysis described in Section 5.2. In reality the amount of human capitalacquired by attending school also depends on such variables as the mental ability of thestudent and the quality of the school that he or she attends. If these omitted variables arepositively correlated with schooling and uncorrelated with the other regressors in thedemand function for health, the schooling coefficient is biased upwards. These biasesare more difficult to sign if, for example, mental ability and school quality are correlatedwith the wage rate.3 3

There are two empirical procedures for assessing whether the investment model givesa more adequate representation of people's behavior than the consumption model. In

33 See Section 7 for a detailed analysis of biases in the regression coefficient of schooling due to the omissionof other variables.

the first place, the wage would have a positive effect on the demand for health in theinvestment model as long as K were less than one. On the other hand, it would have apositive effect in the consumption model only if health were relatively goods-intensive(K < Kz). So, if the computed wage elasticity turns out to be positive, then the largerits value the more likely it is that the investment model is preferable to the consumptionmodel. Of course, provided the production of health were relatively time intensive, thewage elasticity would be negative in the consumption model. In this case, a positiveand statistically significant estimate of Bw would lead to a rejection of the consumptionmodel.

In the second place, health has a zero wealth elasticity in the investment model but apositive wealth elasticity in the consumption model provided it is a superior good. Thissuggests that wealth should be added to the set of regressors in the demand functionsfor health and medical care. Computed wealth elasticities that do not differ significantlyfrom zero would tend to support the investment model.3 4

In addition to estimating demand functions for health and medical care, one couldalso fit the gross investment function given by Equation (44). This would facilitate adirect test of the hypothesis that the more educated are more efficient producers ofhealth. The production function contains two unobserved variables: gross investmentand the own time input. Since, however, -Ht has been assumed to be small relative toSt, one could fit35

In H = Iln M + PH E - t - In o. (49)

The difficulty with the above procedure is that it requires a good estimate of theproduction function. Unfortunately, Equation (49) cannot be fitted by ordinary leastsquares (OLS) because In M and In d0, the disturbance term, are bound to be correlated.From the demand function for medical care

Covariance(ln M, In 60) = (1 - E)Variance(ln 60).

Given < 1, In M and In o would be positively correlated. Since an increase in therate of depreciation lowers the quantity of health capital, the coefficient of medical care

34 Health would have a positive wealth elasticity in the investment model for people who are not in the laborforce. For such individuals, an increase in wealth would raise the ratio of market goods to consumption time,the marginal product of consumption time, and its shadow price. Hence, the monetary rate of return on aninvestment in health would rise. Since my empirical work is limited to members of the labor force, a pureincrease in wealth would not change the shadow price of their time.35 In my monograph, I argue that a one percent increase in medical care would be accompanied by a onepercent increase in own time if factor prices do not vary as more and more health is produced. Therefore,the regression coefficient a in Equation (49) would reflect the sum of the output elasticities of medical careand own time. Given constant returns to scale, the true value of ar should be unity [Grossman (1972b, p. 43)].This analysis becomes much more complicated once joint production is introduced [see Grossman (1972b),Chapter VI and the brief discussion of joint production below].

380 M. Grossman

the first place, the wage would have a positive effect on the demand for health in theinvestment model as long as K were less than one. On the other hand, it would have apositive effect in the consumption model only if health were relatively goods-intensive(K < Kz). So, if the computed wage elasticity turns out to be positive, then the largerits value the more likely it is that the investment model is preferable to the consumptionmodel. Of course, provided the production of health were relatively time intensive, thewage elasticity would be negative in the consumption model. In this case, a positiveand statistically significant estimate of Bw would lead to a rejection of the consumptionmodel.

In the second place, health has a zero wealth elasticity in the investment model but apositive wealth elasticity in the consumption model provided it is a superior good. Thissuggests that wealth should be added to the set of regressors in the demand functionsfor health and medical care. Computed wealth elasticities that do not differ significantlyfrom zero would tend to support the investment model.3 4

In addition to estimating demand functions for health and medical care, one couldalso fit the gross investment function given by Equation (44). This would facilitate adirect test of the hypothesis that the more educated are more efficient producers ofhealth. The production function contains two unobserved variables: gross investmentand the own time input. Since, however, -Ht has been assumed to be small relative toSt, one could fit35

In H = Iln M + PH E - t - In o. (49)

The difficulty with the above procedure is that it requires a good estimate of theproduction function. Unfortunately, Equation (49) cannot be fitted by ordinary leastsquares (OLS) because In M and In d0, the disturbance term, are bound to be correlated.From the demand function for medical care

Covariance(ln M, In 60) = (1 - E)Variance(ln 60).

Given < 1, In M and In o would be positively correlated. Since an increase in therate of depreciation lowers the quantity of health capital, the coefficient of medical care

34 Health would have a positive wealth elasticity in the investment model for people who are not in the laborforce. For such individuals, an increase in wealth would raise the ratio of market goods to consumption time,the marginal product of consumption time, and its shadow price. Hence, the monetary rate of return on aninvestment in health would rise. Since my empirical work is limited to members of the labor force, a pureincrease in wealth would not change the shadow price of their time.35 In my monograph, I argue that a one percent increase in medical care would be accompanied by a onepercent increase in own time if factor prices do not vary as more and more health is produced. Therefore,the regression coefficient a in Equation (49) would reflect the sum of the output elasticities of medical careand own time. Given constant returns to scale, the true value of ar should be unity [Grossman (1972b, p. 43)].This analysis becomes much more complicated once joint production is introduced [see Grossman (1972b),Chapter VI and the brief discussion of joint production below].

380 M. Grossman

would be biased downward. The same bias exists if there are unmeasured determinantsof efficiency in the production of gross investments in health.

The biases inherent in ordinary least squares estimates of health production func-tions were first emphasized by Auster et al. (1969). They have been considered in muchmore detail in the context of infant health by Rosenzweig and Schultz (1983, 1988,1991), Corman et al. (1987), Grossman and Joyce (1990), and Joyce (1994). Consis-tent estimates of the production function can be obtained by two-stage least squares(TSLS). In the present context, wealth, the wage rates, and the price of medical careserve as instruments for medical care. The usefulness of this procedure rests, however,on the validity of the overidentification restrictions and the degree to which the instru-ments explain a significant percentage of the variation in medical care [Bound et al.(1995), Staiger and Stock (1997)]. The TSLS technique is especially problematic whenthe partial effects of several health inputs are desired and when measures of some ofthese inputs are absent. In this situation the overidentification restrictions may not holdbecause wealth and input prices are likely to be correlated with the missing inputs.

In my monograph on the demand for health, I argued that "a production functiontaken by itself tells nothing about producer or consumer behavior, although it does haveimplications for behavior, which operate on the demand curves for health and medicalcare. Thus, they serve to rationalize the forces at work in the reduced form and givethe variables that enter the equations economic significance. Because the reduced formparameters can be used to explain consumer choices and because they can be obtainedby conventional statistical techniques, their interpretation should be pushed as far aspossible. Only then should one resort to a direct estimate of the production function"[Grossman (1972b, p. 44)]. The reader should keep this position in mind in evaluatingmy discussion of the criticism of my model raised by Zweifel and Breyer (1997) inSection 6.

5.2. Data and results

I fitted the equations formulated in Section 5.1 to a nationally representative 1963United States survey conducted by the National Opinion Research Center and the Centerfor Health Administration Studies of the University of Chicago. I measured the stock ofhealth by individuals' self-evaluation of their health status. I measured healthy time, theoutput produced by health capital, either by the complement of the number of restricted-activity days due to illness or injury or the number of work-loss days due to illness orinjury. I measured medical care by personal medical expenditures on doctors, dentists,hospital care, prescribed and nonprescribed drugs, nonmedical practitioners, and med-ical appliances. I had no data on the actual quantities of specific types of services, forexample the number of physician visits. Similarly, I had no data on the prices of theseservices. Thus, I was forced to assume that the price of medical care (P) in the reducedform demand functions either does not vary among consumers or is not correlated withthe other regressors in the demand functions. Neither assumption is likely to be correct

would be biased downward. The same bias exists if there are unmeasured determinantsof efficiency in the production of gross investments in health.

The biases inherent in ordinary least squares estimates of health production func-tions were first emphasized by Auster et al. (1969). They have been considered in muchmore detail in the context of infant health by Rosenzweig and Schultz (1983, 1988,1991), Corman et al. (1987), Grossman and Joyce (1990), and Joyce (1994). Consis-tent estimates of the production function can be obtained by two-stage least squares(TSLS). In the present context, wealth, the wage rates, and the price of medical careserve as instruments for medical care. The usefulness of this procedure rests, however,on the validity of the overidentification restrictions and the degree to which the instru-ments explain a significant percentage of the variation in medical care [Bound et al.(1995), Staiger and Stock (1997)]. The TSLS technique is especially problematic whenthe partial effects of several health inputs are desired and when measures of some ofthese inputs are absent. In this situation the overidentification restrictions may not holdbecause wealth and input prices are likely to be correlated with the missing inputs.

In my monograph on the demand for health, I argued that "a production functiontaken by itself tells nothing about producer or consumer behavior, although it does haveimplications for behavior, which operate on the demand curves for health and medicalcare. Thus, they serve to rationalize the forces at work in the reduced form and givethe variables that enter the equations economic significance. Because the reduced formparameters can be used to explain consumer choices and because they can be obtainedby conventional statistical techniques, their interpretation should be pushed as far aspossible. Only then should one resort to a direct estimate of the production function"[Grossman (1972b, p. 44)]. The reader should keep this position in mind in evaluatingmy discussion of the criticism of my model raised by Zweifel and Breyer (1997) inSection 6.

5.2. Data and results

I fitted the equations formulated in Section 5.1 to a nationally representative 1963United States survey conducted by the National Opinion Research Center and the Centerfor Health Administration Studies of the University of Chicago. I measured the stock ofhealth by individuals' self-evaluation of their health status. I measured healthy time, theoutput produced by health capital, either by the complement of the number of restricted-activity days due to illness or injury or the number of work-loss days due to illness orinjury. I measured medical care by personal medical expenditures on doctors, dentists,hospital care, prescribed and nonprescribed drugs, nonmedical practitioners, and med-ical appliances. I had no data on the actual quantities of specific types of services, forexample the number of physician visits. Similarly, I had no data on the prices of theseservices. Thus, I was forced to assume that the price of medical care (P) in the reducedform demand functions either does not vary among consumers or is not correlated withthe other regressors in the demand functions. Neither assumption is likely to be correct

in light of the well known moral hazard effect of private health insurance.36 The mainindependent variables in the regressions were the age of the individual, the number ofyears of formal schooling he or she completed, his or her weekly wage rate, and familyincome (a proxy for wealth).

The most important regression results in the demand functions are as follows. Educa-tion and the wage rate have positive and statistically significant coefficients in the healthdemand function, regardless of the particular measure of health employed. An increasein age simultaneously reduces health and increases medical expenditures. Both effectsare significant. The signs of the age, wage, and schooling coefficients in the health de-mand function and the sign of the age coefficient in the medical care demand functionare consistent with the predictions contained in the pure investment model.

In the demand function for medical care the wage coefficient is negative but notsignificant, while the schooling coefficient is positive but not significant. The sign of thewage coefficient is not consistent with the pure investment model, and the sign of theschooling coefficient is not consistent with the version of the investment model in whichthe elasticity of the MEC schedule is less than one. In Grossman (1972b, AppendixD), I show that random measurement error in the wage rate and a positive correlationbetween the wage and unmeasured determinants of nonmarket efficiency create biasesthat may explain these results. Other explanations are possible. For example, the wageelasticity of medical care is not necessarily positive in the investment model if waitingand travel time are required to obtain this care [see Equation (33)]. Schooling is likelyto be positively correlated with the generosity of health insurance coverage leading toan upward bias in its estimated effect.

When the production function is estimated by ordinary least squares, the elasticitiesof the three measures of health with respect to medical care are all negative. Presumably,this reflects the strong positive relation between medical care and the depreciation rate.Estimation of the production function by two-stage least squares reverses the sign ofthe medical care elasticity in most cases. The results, however, are sensitive to whetheror not family income is included in the production function as a proxy for missinginputs.

The most surprising finding is that healthy time has a negative family income elas-ticity. If the consumption aspects of health were at all relevant, a literal interpretationof this result is that health is an inferior commodity. That explanation is, however, notconsistent with the positive and significant income elasticity of demand for medicalcare. I offer an alternative explanation based on joint production. Such health inputs ascigarettes, alcohol, and rich food have negative marginal products. If their income elas-ticities exceeded the income elasticities of the beneficial health inputs, the marginal costof gross investment in health would be positively correlated with income. This expla-nation can account for the positive income elasticity of demand for medical care. Given

36 See Zweifel and Manning (2000) for a review of the literature dealing with the effect of health insuranceon the demand for medical care.

382 M. Grossman

in light of the well known moral hazard effect of private health insurance.36 The mainindependent variables in the regressions were the age of the individual, the number ofyears of formal schooling he or she completed, his or her weekly wage rate, and familyincome (a proxy for wealth).

The most important regression results in the demand functions are as follows. Educa-tion and the wage rate have positive and statistically significant coefficients in the healthdemand function, regardless of the particular measure of health employed. An increasein age simultaneously reduces health and increases medical expenditures. Both effectsare significant. The signs of the age, wage, and schooling coefficients in the health de-mand function and the sign of the age coefficient in the medical care demand functionare consistent with the predictions contained in the pure investment model.

In the demand function for medical care the wage coefficient is negative but notsignificant, while the schooling coefficient is positive but not significant. The sign of thewage coefficient is not consistent with the pure investment model, and the sign of theschooling coefficient is not consistent with the version of the investment model in whichthe elasticity of the MEC schedule is less than one. In Grossman (1972b, AppendixD), I show that random measurement error in the wage rate and a positive correlationbetween the wage and unmeasured determinants of nonmarket efficiency create biasesthat may explain these results. Other explanations are possible. For example, the wageelasticity of medical care is not necessarily positive in the investment model if waitingand travel time are required to obtain this care [see Equation (33)]. Schooling is likelyto be positively correlated with the generosity of health insurance coverage leading toan upward bias in its estimated effect.

When the production function is estimated by ordinary least squares, the elasticitiesof the three measures of health with respect to medical care are all negative. Presumably,this reflects the strong positive relation between medical care and the depreciation rate.Estimation of the production function by two-stage least squares reverses the sign ofthe medical care elasticity in most cases. The results, however, are sensitive to whetheror not family income is included in the production function as a proxy for missinginputs.

The most surprising finding is that healthy time has a negative family income elas-ticity. If the consumption aspects of health were at all relevant, a literal interpretationof this result is that health is an inferior commodity. That explanation is, however, notconsistent with the positive and significant income elasticity of demand for medicalcare. I offer an alternative explanation based on joint production. Such health inputs ascigarettes, alcohol, and rich food have negative marginal products. If their income elas-ticities exceeded the income elasticities of the beneficial health inputs, the marginal costof gross investment in health would be positively correlated with income. This expla-nation can account for the positive income elasticity of demand for medical care. Given

36 See Zweifel and Manning (2000) for a review of the literature dealing with the effect of health insuranceon the demand for medical care.

382 M. Grossman

its assumptions, higher income persons simultaneously reduce their demand for healthand increase their demand for medical care if the elasticity of the MEC schedule is lessthan one.

I emphasized in Section 2 that parametric changes in variables that increase healthytime also prolong length of life. Therefore, I also examine variations in age-adjustedmortality rates across states of the United States in 1960. I find a close agreement be-tween mortality and sick time regression coefficients. Increases in schooling or the wagerate lower mortality, while increases in family income raise it.

6. Extensions

In this section I deal with criticisms and empirical and theoretical extensions of myframework. I begin with empirical testing with cross-sectional data by Wagstaff (1986),Erbsland et al. (1995), and Stratmann (1999) in Section 6.1. I pay particular attention toWagstaff's study because it serves as the basis of a criticism of my approach by Zweifeland Breyer (1997), which I also address in Section 6.1. I turn to empirical extensionswith longitudinal data by Van Doorslaer (1987) and Wagstaff (1993) in Section 6.2.These studies introduce costs of adjustment, although in a rather ad hoc manner. I con-sider theoretical developments by Cropper (1977), Muurinen (1982), Dardanoni andWagstaff (1987, 1990), Selden (1993), Chang (1996), and Liljas (1998) in Section 6.3.With the exception of Muurinen's work, these developments all pertain to uncertainty.

6.1. Empirical extensions with cross-sectional data

Wagstaff (1986) uses the 1976 Danish Welfare Survey to estimate a multiple indicatorversion of the structure and reduced form of my demand for health model. He performsa principal components analysis of nineteen measures of non-chronic health problemsto obtain four health indicators that reflect physical mobility, mental health, respira-tory health, and presence of pain. He then uses these four variables as indicators of theunobserved stock of health. His estimation technique is the so-called MIMIC (multi-ple indicators-multiple causes) model developed by Jreskog (1973) and Goldberger(1974) and employs the maximum likelihood procedure contained in J6reskog and S6r-bom (1981). His contribution is unique because it accounts for the multidimensionalnature of good health both at the conceptual level and at the empirical level.

Aside from the MIMIC methodology, there are two principal differences between mywork and Wagstaff's work. First, the structural equation that I obtain is the productionfunction. On the other hand, the structural equation that he obtains is a conditionaloutput demand function. This expresses the quantity demanded of a health input, suchas medical care, as a function of health output, input prices, and exogenous variablesin the production function such as schooling and age. In the context of the structurethat I specified in Section 5.1, the conditional output demand function is obtained bysolving Equation (44) for medical care as a function of health, the own time input,

its assumptions, higher income persons simultaneously reduce their demand for healthand increase their demand for medical care if the elasticity of the MEC schedule is lessthan one.

I emphasized in Section 2 that parametric changes in variables that increase healthytime also prolong length of life. Therefore, I also examine variations in age-adjustedmortality rates across states of the United States in 1960. I find a close agreement be-tween mortality and sick time regression coefficients. Increases in schooling or the wagerate lower mortality, while increases in family income raise it.

6. Extensions

In this section I deal with criticisms and empirical and theoretical extensions of myframework. I begin with empirical testing with cross-sectional data by Wagstaff (1986),Erbsland et al. (1995), and Stratmann (1999) in Section 6.1. I pay particular attention toWagstaff's study because it serves as the basis of a criticism of my approach by Zweifeland Breyer (1997), which I also address in Section 6.1. I turn to empirical extensionswith longitudinal data by Van Doorslaer (1987) and Wagstaff (1993) in Section 6.2.These studies introduce costs of adjustment, although in a rather ad hoc manner. I con-sider theoretical developments by Cropper (1977), Muurinen (1982), Dardanoni andWagstaff (1987, 1990), Selden (1993), Chang (1996), and Liljas (1998) in Section 6.3.With the exception of Muurinen's work, these developments all pertain to uncertainty.

6.1. Empirical extensions with cross-sectional data

Wagstaff (1986) uses the 1976 Danish Welfare Survey to estimate a multiple indicatorversion of the structure and reduced form of my demand for health model. He performsa principal components analysis of nineteen measures of non-chronic health problemsto obtain four health indicators that reflect physical mobility, mental health, respira-tory health, and presence of pain. He then uses these four variables as indicators of theunobserved stock of health. His estimation technique is the so-called MIMIC (multi-ple indicators-multiple causes) model developed by Jreskog (1973) and Goldberger(1974) and employs the maximum likelihood procedure contained in J6reskog and S6r-bom (1981). His contribution is unique because it accounts for the multidimensionalnature of good health both at the conceptual level and at the empirical level.

Aside from the MIMIC methodology, there are two principal differences between mywork and Wagstaff's work. First, the structural equation that I obtain is the productionfunction. On the other hand, the structural equation that he obtains is a conditionaloutput demand function. This expresses the quantity demanded of a health input, suchas medical care, as a function of health output, input prices, and exogenous variablesin the production function such as schooling and age. In the context of the structurethat I specified in Section 5.1, the conditional output demand function is obtained bysolving Equation (44) for medical care as a function of health, the own time input,

schooling, age, and the rate of depreciation in the initial period and then using the cost-minimization condition to replace the own time input with the wage rate and the priceof medical care. Since an increase in the quantity of health demanded increases thedemand for health inputs, the coefficient of health in the conditional demand functionis positive.

Second, Wagstaff utilizes a Frisch (1964) demand function for health in discussingand attempting to estimate the pure consumption model. This is a demand functionin which the marginal utility of lifetime wealth is held constant when the effects ofvariables that alter the marginal cost of investment in health are evaluated. I utilizedit briefly in treating wage effects in the pure consumption model in Section 4 but didnot stress it either theoretically or empirically. The marginal utility of lifetime wealth isnot observed but can be replaced by initial assets and the sum of lifetime wage rates.Since the data are cross-sectional, initial assets and wage rates over the life cycle are notobserved. Wagstaff predicts the missing measures by regressing current assets and thecurrent wage on age, the square of age, and age-invariant socioeconomic characteristics.

Three health inputs are contained in the data: the number of physician visits dur-ing the eight months prior to the survey, the number of weeks spent in a hospital dur-ing the same period, and the number of complaints for which physician-prescribed orself-prescribed medicines were being taken at the time of the interview. To keep mydiscussion of the results manageable, I will focus on the reduced form and conditionaldemand functions for physician visits and on the demand function for health. The readershould keep in mind that the latent variable health obtained from the MIMIC procedureis a positive correlate of good health. Good health is the dependent variable in the re-duced form demand function for health and one of the right-hand side variables in theconditional demand function for physician visits.

Wagstaff estimates his model with and without initial assets and the sum of lifetimewage rates. He terms the former a pure investment model and the latter a pure con-sumption model. Before discussing the results, one conceptual issue should be noted.Wagstaff indicates that medical inputs in Denmark are heavily subsidized and that al-most all of the total cost of gross investment is accounted for by the cost of the own timeinput. He then argues that the wage coefficient should equal zero in the pure investmentdemand function since K, the share of the total cost of gross investment accounted forby time, is equal to one. He also argues that the coefficient of the wage in the demandfunction for medical care should equal one [see the relevant coefficients in Equations(45) and (46)].

Neither of the preceding propositions is necessarily correct. In Section 3 I devel-oped a model in which the price of medical care is zero, but travel and waiting time(q hours to make one physician visit) are required to obtain medical care as well as toproduce health. I also assumed three endogenous inputs in the health production func-tion: M, TH, and a market good whose acquisition does not require time. I then showedthat the wage elasticity of health is positive, while the wage elasticity of medical care isindeterminate in sign [see Equations (32) and (33), both of which hold q constant]. Ifthere are only two inputs and no time required to obtain medical care, the wage elastici-

384 M. Grossman

schooling, age, and the rate of depreciation in the initial period and then using the cost-minimization condition to replace the own time input with the wage rate and the priceof medical care. Since an increase in the quantity of health demanded increases thedemand for health inputs, the coefficient of health in the conditional demand functionis positive.

Second, Wagstaff utilizes a Frisch (1964) demand function for health in discussingand attempting to estimate the pure consumption model. This is a demand functionin which the marginal utility of lifetime wealth is held constant when the effects ofvariables that alter the marginal cost of investment in health are evaluated. I utilizedit briefly in treating wage effects in the pure consumption model in Section 4 but didnot stress it either theoretically or empirically. The marginal utility of lifetime wealth isnot observed but can be replaced by initial assets and the sum of lifetime wage rates.Since the data are cross-sectional, initial assets and wage rates over the life cycle are notobserved. Wagstaff predicts the missing measures by regressing current assets and thecurrent wage on age, the square of age, and age-invariant socioeconomic characteristics.

Three health inputs are contained in the data: the number of physician visits dur-ing the eight months prior to the survey, the number of weeks spent in a hospital dur-ing the same period, and the number of complaints for which physician-prescribed orself-prescribed medicines were being taken at the time of the interview. To keep mydiscussion of the results manageable, I will focus on the reduced form and conditionaldemand functions for physician visits and on the demand function for health. The readershould keep in mind that the latent variable health obtained from the MIMIC procedureis a positive correlate of good health. Good health is the dependent variable in the re-duced form demand function for health and one of the right-hand side variables in theconditional demand function for physician visits.

Wagstaff estimates his model with and without initial assets and the sum of lifetimewage rates. He terms the former a pure investment model and the latter a pure con-sumption model. Before discussing the results, one conceptual issue should be noted.Wagstaff indicates that medical inputs in Denmark are heavily subsidized and that al-most all of the total cost of gross investment is accounted for by the cost of the own timeinput. He then argues that the wage coefficient should equal zero in the pure investmentdemand function since K, the share of the total cost of gross investment accounted forby time, is equal to one. He also argues that the coefficient of the wage in the demandfunction for medical care should equal one [see the relevant coefficients in Equations(45) and (46)].

Neither of the preceding propositions is necessarily correct. In Section 3 I devel-oped a model in which the price of medical care is zero, but travel and waiting time(q hours to make one physician visit) are required to obtain medical care as well as toproduce health. I also assumed three endogenous inputs in the health production func-tion: M, TH, and a market good whose acquisition does not require time. I then showedthat the wage elasticity of health is positive, while the wage elasticity of medical care isindeterminate in sign [see Equations (32) and (33), both of which hold q constant]. Ifthere are only two inputs and no time required to obtain medical care, the wage elastici-

384 M. Grossman

ties of health and medical care are zero. The latter elasticity is zero because the marginalproduct of medical care would be driven to zero if its price is zero for a given wage rate.An increase in the wage rate induces no further substitution in production. With traveland waiting time, the marginal product of care is positive, but the price of medical carerelative to the price of the own time input (Wq/l W) does not depend on W.

An additional complication is that Wagstaff includes a proxy for Wq - the respon-dent's wage multiplied by the time required to travel to his or her physician - in thedemand function for medical care. He asserts that the coefficient of the logarithm ofthis variable should equal the coefficient of the logarithm of the wage in the demandfunction for medical care in a model in which the price of medical care is not zero. Thisis not correct because the logarithm of W is held constant. Hence increases in Wq aredue solely to increases in q. As q rises, H falls and M falls because the price of Mrelative to the price of TH[(P + Wq)/ W] rises. 37

In Wagstaff's estimate of the reduced form of the pure investment model, the wagerate, years of formal schooling completed, and age all have the correct signs and allthree variables are significant in the demand function for health. In the demand functionfor physician visits the schooling variable has a negative and significant coefficient. Thisfinding differs from mine and is in accord with the predictions of the pure investmentmodel. The age coefficient is positive and significant at the 10 percent level on a one-tailed test but not at the 5 percent level. 3 8 The wage coefficient, however, is negative andnot significant. The last finding is consistent with the three-input model outlined aboveand is not necessarily evidence against the investment model.

The time cost variable has the correct negative sign in the demand function for physi-cian visits, but it is not significant. Wagstaff, however, includes the number of physiciansper capita in the respondent's county of residence in the same equation. This variablehas a positive effect on visits, is likely to be negatively related to travel time, and maycapture part of the travel time effect.

Wagstaff concludes the discussion of the results of estimating the reduced form ofthe investment model as follows: "Broadly speaking ... the coefficients are similar tothose reported by Grossman and are consistent with the model's structural parametersbeing of the expected sign. One would seem justified, therefore, in using the ... data forexploring the implications of using structural equation methods in this context" (p. 214).When this is done, the coefficient of good health in the conditional demand function forphysician visits has the wrong sign. It is negative and very significant.

This last finding is used by Zweifel and Breyer (1997) to dismiss my model of thedemand for health. They write: "Unfortunately, empirical evidence consistently fails to

37 The complete specification involves regressing H on W and q and regressing M on W and q, where it isunderstood that all variables are in logarithms. In the three input model in which the money price of medicalcare is zero, the coefficients of q are negative in both equations. The coefficient of W is positive in the healthequation and ambiguous in sign in the medical care equation.38 A one-tailed test is appropriate since the alternative hypothesis is that the age coefficient is positive. Theage coefficient is positive and significant at all conventional levels in the demand functions for hospital staysand medicines.

ties of health and medical care are zero. The latter elasticity is zero because the marginalproduct of medical care would be driven to zero if its price is zero for a given wage rate.An increase in the wage rate induces no further substitution in production. With traveland waiting time, the marginal product of care is positive, but the price of medical carerelative to the price of the own time input (Wq/l W) does not depend on W.

An additional complication is that Wagstaff includes a proxy for Wq - the respon-dent's wage multiplied by the time required to travel to his or her physician - in thedemand function for medical care. He asserts that the coefficient of the logarithm ofthis variable should equal the coefficient of the logarithm of the wage in the demandfunction for medical care in a model in which the price of medical care is not zero. Thisis not correct because the logarithm of W is held constant. Hence increases in Wq aredue solely to increases in q. As q rises, H falls and M falls because the price of Mrelative to the price of TH[(P + Wq)/ W] rises. 37

In Wagstaff's estimate of the reduced form of the pure investment model, the wagerate, years of formal schooling completed, and age all have the correct signs and allthree variables are significant in the demand function for health. In the demand functionfor physician visits the schooling variable has a negative and significant coefficient. Thisfinding differs from mine and is in accord with the predictions of the pure investmentmodel. The age coefficient is positive and significant at the 10 percent level on a one-tailed test but not at the 5 percent level. 3 8 The wage coefficient, however, is negative andnot significant. The last finding is consistent with the three-input model outlined aboveand is not necessarily evidence against the investment model.

The time cost variable has the correct negative sign in the demand function for physi-cian visits, but it is not significant. Wagstaff, however, includes the number of physiciansper capita in the respondent's county of residence in the same equation. This variablehas a positive effect on visits, is likely to be negatively related to travel time, and maycapture part of the travel time effect.

Wagstaff concludes the discussion of the results of estimating the reduced form ofthe investment model as follows: "Broadly speaking ... the coefficients are similar tothose reported by Grossman and are consistent with the model's structural parametersbeing of the expected sign. One would seem justified, therefore, in using the ... data forexploring the implications of using structural equation methods in this context" (p. 214).When this is done, the coefficient of good health in the conditional demand function forphysician visits has the wrong sign. It is negative and very significant.

This last finding is used by Zweifel and Breyer (1997) to dismiss my model of thedemand for health. They write: "Unfortunately, empirical evidence consistently fails to

37 The complete specification involves regressing H on W and q and regressing M on W and q, where it isunderstood that all variables are in logarithms. In the three input model in which the money price of medicalcare is zero, the coefficients of q are negative in both equations. The coefficient of W is positive in the healthequation and ambiguous in sign in the medical care equation.38 A one-tailed test is appropriate since the alternative hypothesis is that the age coefficient is positive. Theage coefficient is positive and significant at all conventional levels in the demand functions for hospital staysand medicines.

M. Grossman

confirm this crucial prediction [that the partial correlation between good health andmedical care should be positive]. When health status is introduced as a latent vari-able through the use of simultaneous indicators, all components of medical care dis-tinguished exhibit a very definite and highly significant negative (their italics) partialrelationship with health .... The notion that expenditure on medical care constitutes ademand derived from an underlying demand for health cannot be upheld because healthstatus and demand for medical care are negatively rather than positively related" (pp.60, 62).

Note, however, that biases arise if the conditional demand function is estimated withhealth treated as exogenous for the same reason that biases arise if the production func-tion is estimated by ordinary least squares. In particular, the depreciation rate in theinitial period (the disturbance term in the equation) is positively correlated with medi-cal care and negatively correlated with health. Hence, the coefficient of health is biaseddownward in the conditional medical care demand function, and this coefficient couldwell be negative. The conditional demand function is much more difficult to estimateby two-stage least squares than the production function because no exogenous variablesare omitted from it in the investment model. Input prices cannot be used to identify thisequation because they are relevant regressors in it. Only wealth and the prices of inputsused to produce commodities other than health are omitted from the conditional de-mand function in the consumption model or in a mixed investment-consumption model.Measures of the latter variables typically are not available.

In his multiple indicator model, Wagstaff does not treat the latent health variable asendogenous when he obtains the conditional demand function. He is careful to point outthat the bias that I have just outlined can explain his result. He also argues that absenceof measures of some inputs may account for his finding. He states: "The identificationof medical care with market inputs in the health investment production function mightbe argued to be a source of potential error. If non-medical inputs are important inputsin the production of health - as clearly they are - one might argue that the results stemfrom a failure to estimate a system (his italics) of structural demand equations for healthinputs" (p. 226). Although I am biased, in my view these considerations go a long waytoward refuting the Zweifel-Breyer critique.

As I indicated above, Wagstaff estimates a reduced form demand function for healthin the context of what he terms a pure consumption model as well as in the context ofa pure investment model. He does this by including initial assets and the lifetime wagevariable in the demand functions as proxies for the marginal utility of wealth. Strictlyspeaking, however, this is not a pure consumption model. It simply accommodates theconsumption motive as well as the investment motive for demanding health. Wagstaffproposes but does not stress one test to distinguish the investment model from the con-sumption model. If the marginal utility of health does not depend on the quantity of theZ-commodity in the current period utility function, the wage effect should be negativein the demand function for health. Empirically, the current wage coefficient remainspositive and significant when initial assets and the lifetime wage are introduced as re-

M. Grossman

confirm this crucial prediction [that the partial correlation between good health andmedical care should be positive]. When health status is introduced as a latent vari-able through the use of simultaneous indicators, all components of medical care dis-tinguished exhibit a very definite and highly significant negative (their italics) partialrelationship with health .... The notion that expenditure on medical care constitutes ademand derived from an underlying demand for health cannot be upheld because healthstatus and demand for medical care are negatively rather than positively related" (pp.60, 62).

Note, however, that biases arise if the conditional demand function is estimated withhealth treated as exogenous for the same reason that biases arise if the production func-tion is estimated by ordinary least squares. In particular, the depreciation rate in theinitial period (the disturbance term in the equation) is positively correlated with medi-cal care and negatively correlated with health. Hence, the coefficient of health is biaseddownward in the conditional medical care demand function, and this coefficient couldwell be negative. The conditional demand function is much more difficult to estimateby two-stage least squares than the production function because no exogenous variablesare omitted from it in the investment model. Input prices cannot be used to identify thisequation because they are relevant regressors in it. Only wealth and the prices of inputsused to produce commodities other than health are omitted from the conditional de-mand function in the consumption model or in a mixed investment-consumption model.Measures of the latter variables typically are not available.

In his multiple indicator model, Wagstaff does not treat the latent health variable asendogenous when he obtains the conditional demand function. He is careful to point outthat the bias that I have just outlined can explain his result. He also argues that absenceof measures of some inputs may account for his finding. He states: "The identificationof medical care with market inputs in the health investment production function mightbe argued to be a source of potential error. If non-medical inputs are important inputsin the production of health - as clearly they are - one might argue that the results stemfrom a failure to estimate a system (his italics) of structural demand equations for healthinputs" (p. 226). Although I am biased, in my view these considerations go a long waytoward refuting the Zweifel-Breyer critique.

As I indicated above, Wagstaff estimates a reduced form demand function for healthin the context of what he terms a pure consumption model as well as in the context ofa pure investment model. He does this by including initial assets and the lifetime wagevariable in the demand functions as proxies for the marginal utility of wealth. Strictlyspeaking, however, this is not a pure consumption model. It simply accommodates theconsumption motive as well as the investment motive for demanding health. Wagstaffproposes but does not stress one test to distinguish the investment model from the con-sumption model. If the marginal utility of health does not depend on the quantity of theZ-commodity in the current period utility function, the wage effect should be negativein the demand function for health. Empirically, the current wage coefficient remainspositive and significant when initial assets and the lifetime wage are introduced as re-

gressors in this equation. As I noted in Section 4, this result also is consistent with apure consumption model in which the marginal utility of H is negatively related to Z.

The initial assets and lifetime wage coefficients are highly significant. As Wagstaffindicates, these variables are highly correlated with schooling, the current wage, andother variables in the demand function for health since both are predicted from thesevariables. These intercorrelations are so high that the schooling coefficient becomesnegative and significant in the consumption demand function. Wagstaff stresses thatthese results must be interpreted with caution.

Zweifel and Breyer (1997) have a confusing and incorrect discussion of theoreticaland empirical results on wage effects. They claim that their discussion, which formspart of the critique of my model, is based on Wagstaff's study. In the demand functionfor health, they indicate that the lifetime wage effect is negative in the pure consumptionmodel and positive in the pure investment model. If the current wage is held constant,both statements are wrong. There is no lifetime wage effect in the investment modeland a positive effect in the consumption model provided that health is a superior com-modity. 39 If their statements pertain to the current wage effect, the sign is positive in theinvestment model and indeterminate in the consumption model whether utility or themarginal utility of wealth is held constant.

Zweifel and Breyer's (1997) discussion of schooling effects can be characterized inthe same manner as their discussion of wage effects. They claim that the consumptionmodel predicts a positive schooling effect in the demand function for health and a neg-ative effect in the demand function for medical care. This is not entirely consistent withthe analysis in Section 4. They use Wagstaff's (1986) result that schooling has a positivecoefficient in the conditional demand function for physician visits as evidence againstmy approach. But as Wagstaff and I have stressed, estimates of that equation are badlybiased because health is not treated as endogenous.

A final criticism made by Zweifel and Breyer is that the wage rate does not ade-quately measure the monetary value of an increase in healthy time due to informal sickleave arrangements and private and social insurance that fund earnings losses due toillness. They do not reconcile this point with the positive effects of the wage rate onvarious health measures in my study and in Wagstaff's study. In addition, sick leaveand insurance plans typically finance less than 100 percent of the loss in earnings. Moreimportantly, they ignore my argument that "... 'the inconvenience costs of illness' arepositively correlated with the wage rate .... The complexity of a particular job andthe amount of responsibility it entails certainly are positively related to the wage. Thus,when an individual with a high wage becomes ill, tasks that only he can perform accu-mulate. These increase the intensity of his work load and give him an incentive to avoidillness by demanding more health capital" [Grossman (1972b, pp. 69-70)].

Erbsland et al. (1995) provide another example of the application of the MIMIC pro-cedure to the estimation of a demand for health model. Their database is the 1986 West

39 Joint production could account for a negative lifetime wage effect, but Zweifel and Breyer do not considerthis phenomenon.

gressors in this equation. As I noted in Section 4, this result also is consistent with apure consumption model in which the marginal utility of H is negatively related to Z.

The initial assets and lifetime wage coefficients are highly significant. As Wagstaffindicates, these variables are highly correlated with schooling, the current wage, andother variables in the demand function for health since both are predicted from thesevariables. These intercorrelations are so high that the schooling coefficient becomesnegative and significant in the consumption demand function. Wagstaff stresses thatthese results must be interpreted with caution.

Zweifel and Breyer (1997) have a confusing and incorrect discussion of theoreticaland empirical results on wage effects. They claim that their discussion, which formspart of the critique of my model, is based on Wagstaff's study. In the demand functionfor health, they indicate that the lifetime wage effect is negative in the pure consumptionmodel and positive in the pure investment model. If the current wage is held constant,both statements are wrong. There is no lifetime wage effect in the investment modeland a positive effect in the consumption model provided that health is a superior com-modity. 39 If their statements pertain to the current wage effect, the sign is positive in theinvestment model and indeterminate in the consumption model whether utility or themarginal utility of wealth is held constant.

Zweifel and Breyer's (1997) discussion of schooling effects can be characterized inthe same manner as their discussion of wage effects. They claim that the consumptionmodel predicts a positive schooling effect in the demand function for health and a neg-ative effect in the demand function for medical care. This is not entirely consistent withthe analysis in Section 4. They use Wagstaff's (1986) result that schooling has a positivecoefficient in the conditional demand function for physician visits as evidence againstmy approach. But as Wagstaff and I have stressed, estimates of that equation are badlybiased because health is not treated as endogenous.

A final criticism made by Zweifel and Breyer is that the wage rate does not ade-quately measure the monetary value of an increase in healthy time due to informal sickleave arrangements and private and social insurance that fund earnings losses due toillness. They do not reconcile this point with the positive effects of the wage rate onvarious health measures in my study and in Wagstaff's study. In addition, sick leaveand insurance plans typically finance less than 100 percent of the loss in earnings. Moreimportantly, they ignore my argument that "... 'the inconvenience costs of illness' arepositively correlated with the wage rate .... The complexity of a particular job andthe amount of responsibility it entails certainly are positively related to the wage. Thus,when an individual with a high wage becomes ill, tasks that only he can perform accu-mulate. These increase the intensity of his work load and give him an incentive to avoidillness by demanding more health capital" [Grossman (1972b, pp. 69-70)].

Erbsland et al. (1995) provide another example of the application of the MIMIC pro-cedure to the estimation of a demand for health model. Their database is the 1986 West

39 Joint production could account for a negative lifetime wage effect, but Zweifel and Breyer do not considerthis phenomenon.

German Socio-economic Panel. The degree of handicap, self-rated health, the durationof sick time, and the number of chronic conditions, all as reported by the individual,serve as four indicators of the unobserved stock of health. In the reduced form demandfunction for health, schooling has a positive and significant coefficient, while age hasa negative and significant coefficient. In the reduced form demand function for visitsto general practitioners, the age effect is positive and significant, while the schoolingeffect is negative and significant. These results are consistent with predictions made bythe investment model. The latent variable health, which is treated as exogenous, has anegative and very significant coefficient in the conditional demand function for physi-cian visits. This is the same finding reported by Wagstaff (1986).

In my 1972 study [Grossman (1972b)], I showed that the sign of the correlation be-tween medical care and health can be reversed if medical care is treated as endogenousin the estimation of health production functions. Stratmann (1999) gives much morerecent evidence in support of the same proposition. Using the 1989 US National HealthInterview Survey, he estimates production functions in which the number of work-lossdays due to illness in the past two weeks serves as the health measure and a dichotomousindicator for a doctor visit in the past two weeks serves as the measure of medical care.In a partial attempt to control for reverse causality from poor health to more medicalcare, he obtains separate production functions for persons with influenza, persons withimpairments, and persons with chronic asthma.

In single equation tobit models, persons who had a doctor visit had significantly morework-loss than persons who did not have a visit for each of the three conditions. In si-multaneous equations probit-tobit models in which the probability of a doctor visit isendogenous, persons who had a doctor visit had significantly less work-loss. The tobitcoefficient in the simultaneous equations model implies that the marginal effect of a doc-tor visit is a 2.7 day reduction in work loss in the case of influenza. 40 The correspondingreductions for impairments and chronic asthma are 2.9 days and 6.9 days, respectively.

6.2. Empirical extensions with longitudinal data

Van Doorslaer (1987) and Wagstaff (1993) fit dynamic demand for health models tolongitudinal data. These efforts potentially are very useful because they allow one totake account of the effects of unmeasured variables such as the rate of depreciationand of reverse causality from health at early stages in the life cycle to the amount offormal schooling completed (see Section 7 for more details). In addition, one can relaxthe assumption that there are no costs of adjustment, so that the lagged stock of healthbecomes a relevant determinant of the current stock of health.

40 Stratmann identifies his model with instruments reflecting the price of visiting a physician which differs

according to the type of health insurance carried by individuals. The specific measures are Medicaid cover-age, private insurance coverage, membership in a Health Maintenance Organization, and whether or not the

employer paid the health insurance premium.

388 M. Grossman

German Socio-economic Panel. The degree of handicap, self-rated health, the durationof sick time, and the number of chronic conditions, all as reported by the individual,serve as four indicators of the unobserved stock of health. In the reduced form demandfunction for health, schooling has a positive and significant coefficient, while age hasa negative and significant coefficient. In the reduced form demand function for visitsto general practitioners, the age effect is positive and significant, while the schoolingeffect is negative and significant. These results are consistent with predictions made bythe investment model. The latent variable health, which is treated as exogenous, has anegative and very significant coefficient in the conditional demand function for physi-cian visits. This is the same finding reported by Wagstaff (1986).

In my 1972 study [Grossman (1972b)], I showed that the sign of the correlation be-tween medical care and health can be reversed if medical care is treated as endogenousin the estimation of health production functions. Stratmann (1999) gives much morerecent evidence in support of the same proposition. Using the 1989 US National HealthInterview Survey, he estimates production functions in which the number of work-lossdays due to illness in the past two weeks serves as the health measure and a dichotomousindicator for a doctor visit in the past two weeks serves as the measure of medical care.In a partial attempt to control for reverse causality from poor health to more medicalcare, he obtains separate production functions for persons with influenza, persons withimpairments, and persons with chronic asthma.

In single equation tobit models, persons who had a doctor visit had significantly morework-loss than persons who did not have a visit for each of the three conditions. In si-multaneous equations probit-tobit models in which the probability of a doctor visit isendogenous, persons who had a doctor visit had significantly less work-loss. The tobitcoefficient in the simultaneous equations model implies that the marginal effect of a doc-tor visit is a 2.7 day reduction in work loss in the case of influenza. 40 The correspondingreductions for impairments and chronic asthma are 2.9 days and 6.9 days, respectively.

6.2. Empirical extensions with longitudinal data

Van Doorslaer (1987) and Wagstaff (1993) fit dynamic demand for health models tolongitudinal data. These efforts potentially are very useful because they allow one totake account of the effects of unmeasured variables such as the rate of depreciationand of reverse causality from health at early stages in the life cycle to the amount offormal schooling completed (see Section 7 for more details). In addition, one can relaxthe assumption that there are no costs of adjustment, so that the lagged stock of healthbecomes a relevant determinant of the current stock of health.

40 Stratmann identifies his model with instruments reflecting the price of visiting a physician which differs

according to the type of health insurance carried by individuals. The specific measures are Medicaid cover-age, private insurance coverage, membership in a Health Maintenance Organization, and whether or not the

employer paid the health insurance premium.

388 M. Grossman

Van Doorslaer (1987) employs the 1984 Netherlands Health Interview Survey. Whilethis is a cross-sectional survey, respondents were asked to evaluate their health in 1979as well as in 1984. Both measures are ten-point scales, where the lowest category isvery poor health and the highest category is very good health.

Van Doorslaer uses the identity that the current stock of health equals the undepreci-ated component of the past stock plus gross investment:

Ht = (1 -- t_1)Ht_- + It-,. (50)

He assumes that gross investment is a function of personal background variables(schooling, age, income, and gender). Thus, he regresses health in 1984 on these vari-ables and on health in 1979. To test Muurinen's (1982) hypothesis that schooling lowersthe rate of depreciation (see Section 4.2), he allows for an interaction between this vari-able and health in 1979 in some of the estimated models.

Van Doorslaer's main finding is that schooling has a positive and significant coef-ficient in the regression explaining health in 1984, with health in 1979 held constant.The regressions in which schooling, past health, and an interaction between the two areentered as regressors are plagued by multicollinearity. They do not allow one to distin-guish Muurinen's hypothesis from the hypothesis that schooling raises efficiency in theproduction of health.

Wagstaff (1993) uses the Danish Health Study, which followed respondents over aperiod of 12 months beginning in October 1982. As in his 1986 study, a MIMIC modelis estimated. Three health measures are used as indicators of the unobserved stock ofhealth capital in 1982 (past stock) and 1983 (current stock). These are a dichotomousindicator of the presence of a health limitation, physician-assessed health of the respon-dent as reported by the respondent, and self-assessed health.4 1 Both of the assessmentvariables have five-point scales. Unlike his 1986 study, Wagstaff also treats gross in-vestment in health as a latent variable. There are six health care utilization indicatorsof gross investment: the number of consultations with a general practitioner over theyear, the number of consultations with a specialist over the year, the number of days asan inpatient in a hospital over the year, the number of sessions with a physiotherapistover the year, the number of hospital outpatient visits over the year, and the number ofhospital emergency room visits during the year.

Wagstaff explicitly assumes partial adjustment instead of instantaneous adjustment.He also assumes that the reduced form demand for health equation is linear rather thanlog-linear. He argues that this makes it compatible with the linear nature of the netinvestment identity (net investment equals gross investment minus depreciation). The

41 The health limitation variable for 1982 is based on responses during October, November, and Decemberof that year. The corresponding variable for 1983 is based on responses during the months of January throughSeptember. It is not clear which month is used for the self- and physician-rated health measures. I assumethat the 1982 measures come from the October 1982 survey and the 1983 measure comes from the September1983 survey.

Van Doorslaer (1987) employs the 1984 Netherlands Health Interview Survey. Whilethis is a cross-sectional survey, respondents were asked to evaluate their health in 1979as well as in 1984. Both measures are ten-point scales, where the lowest category isvery poor health and the highest category is very good health.

Van Doorslaer uses the identity that the current stock of health equals the undepreci-ated component of the past stock plus gross investment:

Ht = (1 -- t_1)Ht_- + It-,. (50)

He assumes that gross investment is a function of personal background variables(schooling, age, income, and gender). Thus, he regresses health in 1984 on these vari-ables and on health in 1979. To test Muurinen's (1982) hypothesis that schooling lowersthe rate of depreciation (see Section 4.2), he allows for an interaction between this vari-able and health in 1979 in some of the estimated models.

Van Doorslaer's main finding is that schooling has a positive and significant coef-ficient in the regression explaining health in 1984, with health in 1979 held constant.The regressions in which schooling, past health, and an interaction between the two areentered as regressors are plagued by multicollinearity. They do not allow one to distin-guish Muurinen's hypothesis from the hypothesis that schooling raises efficiency in theproduction of health.

Wagstaff (1993) uses the Danish Health Study, which followed respondents over aperiod of 12 months beginning in October 1982. As in his 1986 study, a MIMIC modelis estimated. Three health measures are used as indicators of the unobserved stock ofhealth capital in 1982 (past stock) and 1983 (current stock). These are a dichotomousindicator of the presence of a health limitation, physician-assessed health of the respon-dent as reported by the respondent, and self-assessed health.4 1 Both of the assessmentvariables have five-point scales. Unlike his 1986 study, Wagstaff also treats gross in-vestment in health as a latent variable. There are six health care utilization indicatorsof gross investment: the number of consultations with a general practitioner over theyear, the number of consultations with a specialist over the year, the number of days asan inpatient in a hospital over the year, the number of sessions with a physiotherapistover the year, the number of hospital outpatient visits over the year, and the number ofhospital emergency room visits during the year.

Wagstaff explicitly assumes partial adjustment instead of instantaneous adjustment.He also assumes that the reduced form demand for health equation is linear rather thanlog-linear. He argues that this makes it compatible with the linear nature of the netinvestment identity (net investment equals gross investment minus depreciation). The

41 The health limitation variable for 1982 is based on responses during October, November, and Decemberof that year. The corresponding variable for 1983 is based on responses during the months of January throughSeptember. It is not clear which month is used for the self- and physician-rated health measures. I assumethat the 1982 measures come from the October 1982 survey and the 1983 measure comes from the September1983 survey.

desired stock in period t is a linear function of age, schooling, family income, andgender. A fraction (t) of the gap between the actual and the desired stock is closedeach period. Hence the lagged stock enters the reduced form demand function with acoefficient equal to I - t. Solving Equation (50) for gross investment in period t - 1 andreplacing Ht by its demand function, Wagstaff obtains a demand function for It 1I thatdepends on the same variables as those in the demand function for Ht. The coefficientof each sociodemographic variable in the demand function for Ht is the same as thecorresponding coefficient in the demand function for 11 _ 1. The coefficient on the laggedstock in the latter demand function equals - (/ - t- 1). By estimating the model withcross-equation constraints, tt and St-l are identified.

Wagstaff emphasizes that the same variables enter his conditional demand functionfor It-, in his cost-of-adjustment model as those that enter the conditional demandfunction for Itl in my instantaneous adjustment model. The interpretation of the pa-rameters, however, differs. In my case, the contemporaneous health stock has a positivecoefficient, whereas in his case the coefficient is negative if , exceeds t- 1. In my case,the coefficient of schooling, for example, is equal to the negative of the schooling co-efficient in the production function. In his case, it equals the coefficient of schooling inthe demand function for H.

To allow for the possibility that the rate of depreciation varies with age, Wagstaff fitsthe model separately for adults under the age of forty-one and for adults greater thanor equal to this age. For each age group, schooling has a positive and significant effecton current health with past health held constant. The coefficient of Hi_1 in the demandfunction for Itl is negative, suggesting costs of adjustment and also suggesting thatAt exceeds 5t- 1. The implied value of the rate of depreciation is, however, larger in thesample of younger adults than in the sample of older adults. Moreover, in the lattersample, the estimated rate of depreciation is negative. These implausible findings maybe traced to the inordinate demands on the data attributed to the MIMIC methodologywith two latent variables and cross-equation constraints.

Some conceptual issues can be raised in evaluating the two studies just discussed.Wagstaff (1993) estimates input demand functions which include availability measuresas proxies for travel and waiting time (for example, the per capita number of generalpractitioners in the individual's district in the demand function for the number of con-sultations with general practitioners). Yet he excludes these variables from the demandfunctions for Ht and Itl. This is not justified. In addition, Wagstaff implies that thegross investment production function is linear in its inputs, which violates the cost-minimization conditions.

More fundamentally, both Van Doorslaer (1987) and Wagstaff (1993) provide ad hoccost-of-adjustment models. I now show that a rigorous development of such a modelcontains somewhat different demand functions than the ones that they estimate. To sim-plify, I assume that the pure investment model is valid, ignore complications with cost-of-adjustment models studied by Ehrlich and Chuma (1990), and fix the wage rate at

390 M. Grossman

desired stock in period t is a linear function of age, schooling, family income, andgender. A fraction (t) of the gap between the actual and the desired stock is closedeach period. Hence the lagged stock enters the reduced form demand function with acoefficient equal to I - t. Solving Equation (50) for gross investment in period t - 1 andreplacing Ht by its demand function, Wagstaff obtains a demand function for It 1I thatdepends on the same variables as those in the demand function for Ht. The coefficientof each sociodemographic variable in the demand function for Ht is the same as thecorresponding coefficient in the demand function for 11 _ 1. The coefficient on the laggedstock in the latter demand function equals - (/ - t- 1). By estimating the model withcross-equation constraints, tt and St-l are identified.

Wagstaff emphasizes that the same variables enter his conditional demand functionfor It-, in his cost-of-adjustment model as those that enter the conditional demandfunction for Itl in my instantaneous adjustment model. The interpretation of the pa-rameters, however, differs. In my case, the contemporaneous health stock has a positivecoefficient, whereas in his case the coefficient is negative if , exceeds t- 1. In my case,the coefficient of schooling, for example, is equal to the negative of the schooling co-efficient in the production function. In his case, it equals the coefficient of schooling inthe demand function for H.

To allow for the possibility that the rate of depreciation varies with age, Wagstaff fitsthe model separately for adults under the age of forty-one and for adults greater thanor equal to this age. For each age group, schooling has a positive and significant effecton current health with past health held constant. The coefficient of Hi_1 in the demandfunction for Itl is negative, suggesting costs of adjustment and also suggesting thatAt exceeds 5t- 1. The implied value of the rate of depreciation is, however, larger in thesample of younger adults than in the sample of older adults. Moreover, in the lattersample, the estimated rate of depreciation is negative. These implausible findings maybe traced to the inordinate demands on the data attributed to the MIMIC methodologywith two latent variables and cross-equation constraints.

Some conceptual issues can be raised in evaluating the two studies just discussed.Wagstaff (1993) estimates input demand functions which include availability measuresas proxies for travel and waiting time (for example, the per capita number of generalpractitioners in the individual's district in the demand function for the number of con-sultations with general practitioners). Yet he excludes these variables from the demandfunctions for Ht and Itl. This is not justified. In addition, Wagstaff implies that thegross investment production function is linear in its inputs, which violates the cost-minimization conditions.

More fundamentally, both Van Doorslaer (1987) and Wagstaff (1993) provide ad hoccost-of-adjustment models. I now show that a rigorous development of such a modelcontains somewhat different demand functions than the ones that they estimate. To sim-plify, I assume that the pure investment model is valid, ignore complications with cost-of-adjustment models studied by Ehrlich and Chuma (1990), and fix the wage rate at

390 M. Grossman

$1. I also make use of the exact form of the first-order condition for Ht in a discretetime model (see footnote 13):

Gt = (1 + r)rt-1 - (1 - t)7ct. (51)

Note that rt_ is the marginal cost of gross investment in health. Since marginal costrises as the quantity of investment rises in a model with costs of adjustment, the marginalcost of investment exceeds the average cost of investment. Also to simplify and to keepthe system linear, I assume that Gt is a linear function of Ht and that rt-l is a lin-ear function of It-i and Pt-l (the price of the single market input used in the grossinvestment production function):

Gt = p - aHt, (52)

7rt-1 = Pt-, + It-. (53)

Given this model, the optimal stock of health in period t is

(p + r) (1 + r) (1 - t) (1 - t)Ht = Pt-I- It- + Pt+ It. (54)

a al a a a

Since It-l = Ht- (1 - 8t-_)Ht-_ and It(I - St) = (1 - )Ht+l - (1 - St)2Ht,

PH (1 + r) ( +r)( - t-

+ Ht+ + Pt (55)D D

where D = a + (1 + r) + (1 - t)2. Alternatively, substitute Equation (55) into thedefinition of It- I to obtain

I t - (1 + r) - (1 - 6t-)[ + (1 - t)2] HD D D

O - S(1 - t)+ ( H+ Pt (56)

Equation (55) is the demand for health function obtained by Van Doorslaer (1987)and Wagstaff (1993). Their estimates are biased because the stock of health in periodt depends on the stock of health in period t + 1 as well as on the stock of health inperiod t - 1 in a model with costs of adjustment. Equation (56) is the gross investmentdemand function obtained by Wagstaff. His estimate is biased because gross investmentin period t - 1 depends on the stock of health in period t + 1 as well as on the stockof health in period t - 1. Note that this equation and Equation (55) also depend onmeasured and unmeasured determinants of market and nonmarket efficiency in periodst - 1 and t.

$1. I also make use of the exact form of the first-order condition for Ht in a discretetime model (see footnote 13):

Gt = (1 + r)rt-1 - (1 - t)7ct. (51)

Note that rt_ is the marginal cost of gross investment in health. Since marginal costrises as the quantity of investment rises in a model with costs of adjustment, the marginalcost of investment exceeds the average cost of investment. Also to simplify and to keepthe system linear, I assume that Gt is a linear function of Ht and that rt-l is a lin-ear function of It-i and Pt-l (the price of the single market input used in the grossinvestment production function):

Gt = p - aHt, (52)

7rt-1 = Pt-, + It-. (53)

Given this model, the optimal stock of health in period t is

(p + r) (1 + r) (1 - t) (1 - t)Ht = Pt-I- It- + Pt+ It. (54)

a al a a a

Since It-l = Ht- (1 - 8t-_)Ht-_ and It(I - St) = (1 - )Ht+l - (1 - St)2Ht,

PH (1 + r) ( +r)( - t-

+ Ht+ + Pt (55)D D

where D = a + (1 + r) + (1 - t)2. Alternatively, substitute Equation (55) into thedefinition of It- I to obtain

I t - (1 + r) - (1 - 6t-)[ + (1 - t)2] HD D D

O - S(1 - t)+ ( H+ Pt (56)

Equation (55) is the demand for health function obtained by Van Doorslaer (1987)and Wagstaff (1993). Their estimates are biased because the stock of health in periodt depends on the stock of health in period t + 1 as well as on the stock of health inperiod t - 1 in a model with costs of adjustment. Equation (56) is the gross investmentdemand function obtained by Wagstaff. His estimate is biased because gross investmentin period t - 1 depends on the stock of health in period t + 1 as well as on the stockof health in period t - 1. Note that this equation and Equation (55) also depend onmeasured and unmeasured determinants of market and nonmarket efficiency in periodst - 1 and t.

The second-order difference equations given by (55) and (56) can be solved to ex-press Ht or It-I as functions of current, past, and future values of all the exogenousvariables. Similarly, Ht_ and Ht+l depend on this set of exogenous variables. Sinceone of the members of this set is the disturbance term in (55) or (56), the lagged andfuture stocks are correlated with the regression disturbance. Consequently, biases ariseif either equation is estimated by ordinary least squares. Consistent estimates can beobtained by fitting the equations by two-stage least squares with past and future valuesof the exogenous variables serving as instruments for the one-period lead and the one-period lag of the stock.42 Note that consistent estimates cannot be obtained by the ap-plication of ordinary least squares to a first-difference model or to a fixed-effects model.Lagged and future health variables do not drop out of these models and are correlatedwith the time-varying component of the disturbance term.

I conclude that cost-of-adjustment models require at least three data points (threeobservations on each individual) to be estimated. Calculated parameters of this modelare biased if they are obtained by ordinary least squares. There is an added complicationthat arises even if all the necessary data are available because the procedure that I haveoutlined assumes that individuals have perfect information about the future values ofthe exogenous variables. This may or may not be the case.4 3 While the two studies thatI have reviewed are provocative, they do not contain enough information to compareinstantaneous-adjustment models to cost-of-adjustment models.

6.3. Theoretical extensions

Muurinen (1982) examines comparative static age, schooling, and wealth effects in thecontext of a mixed investment-consumption model with perfect certainty. This approachis more general than mine because it incorporates both the investment motive and theconsumption motive for demanding health. In deriving formulas for the effects of in-creases in age and schooling on the optimal quantities of health capital and medicalcare, Muurinen assumes that the undiscounted monetary value of the marginal utilityof healthy time in period t, given by (Uht/X)(l + r)t, is constant for all t. If mt isthe marginal cost of the Z commodity and Ut is its marginal utility, the undiscountedmonetary value of the marginal utility of healthy time also is given by (Uht/Ut)mt.Hence, Muurinen is assuming that the marginal rate of substitution between healthytime and the Z commodity is constant or that the two commodities are perfect substi-tutes. Clearly, this is a very restrictive assumption. 44

42 The minimum requirements for instruments are measures of Pt+, and Pt-2. The one-period lead of theprice has no impact on Ht with Ht+1 held constant. Similarly, the two-period lag of the price has no impacton Ht with Ht I held constant.43 The same issue arises in estimating the rational addiction model of consumer behavior. For a detaileddiscussion, see Becker et al. (1994).44 Ried (1998) also develops a framework for examining the impacts of changes in exogenous variables inthe context of a mixed investment-consumption model. He uses Frisch (1964) demand curves to decompose

392 M. Grossman

The second-order difference equations given by (55) and (56) can be solved to ex-press Ht or It-I as functions of current, past, and future values of all the exogenousvariables. Similarly, Ht_ and Ht+l depend on this set of exogenous variables. Sinceone of the members of this set is the disturbance term in (55) or (56), the lagged andfuture stocks are correlated with the regression disturbance. Consequently, biases ariseif either equation is estimated by ordinary least squares. Consistent estimates can beobtained by fitting the equations by two-stage least squares with past and future valuesof the exogenous variables serving as instruments for the one-period lead and the one-period lag of the stock.42 Note that consistent estimates cannot be obtained by the ap-plication of ordinary least squares to a first-difference model or to a fixed-effects model.Lagged and future health variables do not drop out of these models and are correlatedwith the time-varying component of the disturbance term.

I conclude that cost-of-adjustment models require at least three data points (threeobservations on each individual) to be estimated. Calculated parameters of this modelare biased if they are obtained by ordinary least squares. There is an added complicationthat arises even if all the necessary data are available because the procedure that I haveoutlined assumes that individuals have perfect information about the future values ofthe exogenous variables. This may or may not be the case.4 3 While the two studies thatI have reviewed are provocative, they do not contain enough information to compareinstantaneous-adjustment models to cost-of-adjustment models.

6.3. Theoretical extensions

Muurinen (1982) examines comparative static age, schooling, and wealth effects in thecontext of a mixed investment-consumption model with perfect certainty. This approachis more general than mine because it incorporates both the investment motive and theconsumption motive for demanding health. In deriving formulas for the effects of in-creases in age and schooling on the optimal quantities of health capital and medicalcare, Muurinen assumes that the undiscounted monetary value of the marginal utilityof healthy time in period t, given by (Uht/X)(l + r)t, is constant for all t. If mt isthe marginal cost of the Z commodity and Ut is its marginal utility, the undiscountedmonetary value of the marginal utility of healthy time also is given by (Uht/Ut)mt.Hence, Muurinen is assuming that the marginal rate of substitution between healthytime and the Z commodity is constant or that the two commodities are perfect substi-tutes. Clearly, this is a very restrictive assumption. 44

42 The minimum requirements for instruments are measures of Pt+, and Pt-2. The one-period lead of theprice has no impact on Ht with Ht+1 held constant. Similarly, the two-period lag of the price has no impacton Ht with Ht I held constant.43 The same issue arises in estimating the rational addiction model of consumer behavior. For a detaileddiscussion, see Becker et al. (1994).44 Ried (1998) also develops a framework for examining the impacts of changes in exogenous variables inthe context of a mixed investment-consumption model. He uses Frisch (1964) demand curves to decompose

392 M. Grossman

In my formal development of the demand for health, I ruled out uncertainty. Surelythat is not realistic. I briefly indicated that one could introduce this phenomenon byassuming that a given consumer faces a probability distribution of depreciation rates inevery period. I speculated, but did not prove, that consumers might have incentives tohold an excess stock of health in relatively desirable "states of the world" (outcomeswith relatively low depreciation rates) in order to reduce the loss associated with anunfavorable outcome. In these relatively desirable states, the marginal monetary returnon an investment in health might be smaller than the opportunity cost of capital in apure investment model [Grossman (1972b, pp. 19-21)].

Beginning with Cropper (1977), a number of persons formally have introduced un-certainty into my pure investment model. Cropper assumes that illness occurs in a givenperiod if the stock of health falls below a critical sickness level, which is random. In-come is zero in the illness state. An increase in the stock of health lowers the probabilityof this state. Cropper further assumes that savings are not possible (all income takes theform of earnings) and that consumers are risk-neutral in the sense that their objective isto maximize the expected discounted value of lifetime wealth.4 5

In my view, Cropper's main result is that consumers with higher incomes or wealthlevels will maintain higher stocks of health than poorer persons. While this may appearto be a different result than that contained in my pure investment model with perfect cer-tainty, it is not for two reasons. First, an increase in the stock of health lowers the prob-ability of illness but has no impact on earnings in non-illness states. Hence the marginalbenefit of an increase in the stock is given by the reduction in the probability of illnessmultiplied by the difference between income and gross investment outlays. With theseoutlays held constant, an increase in income raises the marginal benefit and the marginalrate of return on an investment. 46 Therefore, this wealth or income effect is analogousto the wage effect in my pure investment model with perfect certainty. Second, considera pure investment model with perfect certainty, positive initial assets but no possibilityto save or borrow in financial markets. In this model, investment in health is the onlymechanism to increase future consumption. An increase in initial assets will increasethe optimal stock of health provided future consumption has a positive wealth elasticity.

Later treatments of uncertainty in the context of demand for health models have as-sumed risk-averse behavior, so that an expected utility function that exhibits diminish-ing marginal utility of present and future consumption is maximized. Dardanoni andWagstaff (1987), Selden (1993), and Chang (1996) all employ two-period models in

the total effect into an effect that holds the marginal utility of wealth constant and an effect attributable toa change in the marginal utility of wealth. He obtains few, if any, unambiguous predictions. I leave it to thereader to evaluate this contribution.45 Cropper begins with a model in which consumers are risk-averse, but the rate of depreciation on the stockof health does not depend on age. She introduces risk-neutrality when she allows the rate of depreciation todepend on age.46 Cropper assumes that gross investment in health is produced only with medical care, but the above resultholds as long as the share of the own time input in the total cost of gross investment is less than one.

In my formal development of the demand for health, I ruled out uncertainty. Surelythat is not realistic. I briefly indicated that one could introduce this phenomenon byassuming that a given consumer faces a probability distribution of depreciation rates inevery period. I speculated, but did not prove, that consumers might have incentives tohold an excess stock of health in relatively desirable "states of the world" (outcomeswith relatively low depreciation rates) in order to reduce the loss associated with anunfavorable outcome. In these relatively desirable states, the marginal monetary returnon an investment in health might be smaller than the opportunity cost of capital in apure investment model [Grossman (1972b, pp. 19-21)].

Beginning with Cropper (1977), a number of persons formally have introduced un-certainty into my pure investment model. Cropper assumes that illness occurs in a givenperiod if the stock of health falls below a critical sickness level, which is random. In-come is zero in the illness state. An increase in the stock of health lowers the probabilityof this state. Cropper further assumes that savings are not possible (all income takes theform of earnings) and that consumers are risk-neutral in the sense that their objective isto maximize the expected discounted value of lifetime wealth.4 5

In my view, Cropper's main result is that consumers with higher incomes or wealthlevels will maintain higher stocks of health than poorer persons. While this may appearto be a different result than that contained in my pure investment model with perfect cer-tainty, it is not for two reasons. First, an increase in the stock of health lowers the prob-ability of illness but has no impact on earnings in non-illness states. Hence the marginalbenefit of an increase in the stock is given by the reduction in the probability of illnessmultiplied by the difference between income and gross investment outlays. With theseoutlays held constant, an increase in income raises the marginal benefit and the marginalrate of return on an investment. 46 Therefore, this wealth or income effect is analogousto the wage effect in my pure investment model with perfect certainty. Second, considera pure investment model with perfect certainty, positive initial assets but no possibilityto save or borrow in financial markets. In this model, investment in health is the onlymechanism to increase future consumption. An increase in initial assets will increasethe optimal stock of health provided future consumption has a positive wealth elasticity.

Later treatments of uncertainty in the context of demand for health models have as-sumed risk-averse behavior, so that an expected utility function that exhibits diminish-ing marginal utility of present and future consumption is maximized. Dardanoni andWagstaff (1987), Selden (1993), and Chang (1996) all employ two-period models in

the total effect into an effect that holds the marginal utility of wealth constant and an effect attributable toa change in the marginal utility of wealth. He obtains few, if any, unambiguous predictions. I leave it to thereader to evaluate this contribution.45 Cropper begins with a model in which consumers are risk-averse, but the rate of depreciation on the stockof health does not depend on age. She introduces risk-neutrality when she allows the rate of depreciation todepend on age.46 Cropper assumes that gross investment in health is produced only with medical care, but the above resultholds as long as the share of the own time input in the total cost of gross investment is less than one.

which the current period utility function depends only on current consumption. Uncer-tainty in the second period arises because the earnings-generating function in that periodcontains a random variable. This function is Y2 = Wh2(H2 , R) = F(H2, R), where Y2is earnings in period two, h2 is the amount of healthy time in that period, H2 is the stockof health, and R is the random term. Clearly, F1 > 0 and F1 < 0, where F and Fl 1 arethe first and second derivatives of H2 in the earnings function. The second derivativeis negative because of my assumption that the marginal product of the stock of healthin the production of healthy time falls as the stock rises. An increase in R raises earn-ings (F2 > 0). In addition to income or earnings from health, income is available fromsavings at a fixed rate of return.

Given uncertainty, risk-averse individuals make larger investments in health than theywould in its absence. Indeed, the expected marginal rate of return is smaller than the ratewith perfect certainty. This essentially confirms a result that I anticipated in the briefdiscussion of uncertainty in my monograph.

The main impact of the introduction of uncertainty is that the quantities of healthcapital and gross investment depend on initial assets, with the wage rate held constant.The direction of these effects, however, is ambiguous because it depends on the way inwhich risk is specified. Dardanoni and Wagstaff (1987) adopt a multiplicative specifica-tion in which the earnings function is Y2 = RH2 . They show that an increase in initialassets raises the optimal quantities of health and medical care if the utility function ex-hibits decreasing absolute risk aversion.4 7 Selden (1993) adopts a linear specification inwhich the earnings function is Y2 = F(H2 + R) and a2Y2 /a(H2 + R)2 < 0. He reachesthe opposite conclusion: health and medical care fall as assets rise given declining ab-solute risk aversion.

Chang (1996) generalizes the specification of risk. He shows that the sign of the asseteffect depends on the sign of the second-order cross partial derivative in the earningsfunction (F1 2). If F12 is positive and F is zero, the asset effect is positive. This isthe case considered by Dardanoni and Wagstaff. In my view it is not realistic becauseit assumes that the marginal product of the stock of health in the production of healthytime is constant. Given the more realistic case in which F 1I is negative, the asset effect isnegative if Fl 2 is negative (Selden's case) and indeterminate in sign if F12 is positive. 48

Dardanoni and Wagstaff (1990) introduce uncertainty into pure consumption modelsof the demand for health. Their study is a static one-period model. The utility func-tion depends on the consumption of a composite commodity and health. Health is givenby H = R + I(M, R), where R is a random variable and I(M, R) denotes the healthproduction function. They consider two models: one in which H = R + M and onein which H = RM. In the first model an increase in the variance of R with the meanheld constant increases the quantity of medical care demanded under plausible assump-tions about the utility function (superiority of the composite consumption good and

47 A utility function U = U(C) exhibits decreasing absolute risk aversion if -Ucc/ UC falls as C rises.48 Chang provides more results by assuming that the earnings function depends only on "post-shock health,"f(H 2 , R). The finding that the sign of the asset effect is ambiguous holds in his formulation.

394 M. Grossman

which the current period utility function depends only on current consumption. Uncer-tainty in the second period arises because the earnings-generating function in that periodcontains a random variable. This function is Y2 = Wh2(H2 , R) = F(H2, R), where Y2is earnings in period two, h2 is the amount of healthy time in that period, H2 is the stockof health, and R is the random term. Clearly, F1 > 0 and F1 < 0, where F and Fl 1 arethe first and second derivatives of H2 in the earnings function. The second derivativeis negative because of my assumption that the marginal product of the stock of healthin the production of healthy time falls as the stock rises. An increase in R raises earn-ings (F2 > 0). In addition to income or earnings from health, income is available fromsavings at a fixed rate of return.

Given uncertainty, risk-averse individuals make larger investments in health than theywould in its absence. Indeed, the expected marginal rate of return is smaller than the ratewith perfect certainty. This essentially confirms a result that I anticipated in the briefdiscussion of uncertainty in my monograph.

The main impact of the introduction of uncertainty is that the quantities of healthcapital and gross investment depend on initial assets, with the wage rate held constant.The direction of these effects, however, is ambiguous because it depends on the way inwhich risk is specified. Dardanoni and Wagstaff (1987) adopt a multiplicative specifica-tion in which the earnings function is Y2 = RH2 . They show that an increase in initialassets raises the optimal quantities of health and medical care if the utility function ex-hibits decreasing absolute risk aversion.4 7 Selden (1993) adopts a linear specification inwhich the earnings function is Y2 = F(H2 + R) and a2Y2 /a(H2 + R)2 < 0. He reachesthe opposite conclusion: health and medical care fall as assets rise given declining ab-solute risk aversion.

Chang (1996) generalizes the specification of risk. He shows that the sign of the asseteffect depends on the sign of the second-order cross partial derivative in the earningsfunction (F1 2). If F12 is positive and F is zero, the asset effect is positive. This isthe case considered by Dardanoni and Wagstaff. In my view it is not realistic becauseit assumes that the marginal product of the stock of health in the production of healthytime is constant. Given the more realistic case in which F 1I is negative, the asset effect isnegative if Fl 2 is negative (Selden's case) and indeterminate in sign if F12 is positive. 48

Dardanoni and Wagstaff (1990) introduce uncertainty into pure consumption modelsof the demand for health. Their study is a static one-period model. The utility func-tion depends on the consumption of a composite commodity and health. Health is givenby H = R + I(M, R), where R is a random variable and I(M, R) denotes the healthproduction function. They consider two models: one in which H = R + M and onein which H = RM. In the first model an increase in the variance of R with the meanheld constant increases the quantity of medical care demanded under plausible assump-tions about the utility function (superiority of the composite consumption good and

47 A utility function U = U(C) exhibits decreasing absolute risk aversion if -Ucc/ UC falls as C rises.48 Chang provides more results by assuming that the earnings function depends only on "post-shock health,"f(H 2 , R). The finding that the sign of the asset effect is ambiguous holds in his formulation.

394 M. Grossman

non-increasing absolute risk aversion with regard to that good). In the second model thesame effect is more difficult to sign, although it is positive if an increase in R leads to areduction in M and an increase in the composite good and if the utility function exhibitsnon-increasing relative risk aversion with regard to the composite good.49

Liljas (1998) considers uncertainty in the context of a multiperiod mixed investment-consumption model. Uncertainty takes the form of a random variable that affects thestock of health in period t in an additive fashion. He shows that the stock of health islarger in the stochastic case than in the certainty case. Presumably, this result pertainsto the expected stock of health in period t. The actual stock should be smaller than thestock with certainty given a negative shock. Social insurance that funds part of the lossin income due to illness lowers the optimal stock. Private insurance that also funds partof this loss will not necessarily lower the stock further and may actually increase it ifthe cost of this insurance falls as the stock rises.

To summarize, compared to a model with perfect certainty, the expected value of thestock of health is larger and the optimal quantities of gross investment and health inputsalso are larger in a model with uncertainty. In a pure investment model an increase ininitial assets can cause health and medical care to change, but the direction of theseeffects is ambiguous. Under reasonable assumptions, an increase in the variance of riskraises optimal medical care in a pure consumption model.

How valuable are these results? With the exception of the ambiguity of the asseteffect, they are not very surprising. The variance of risk is extremely difficult to measure.I am not aware of empirical studies that have attempted to include this variable in ademand for health framework. The possibility that the asset effect can be nonzero in apure investment model provides an alternative explanation of Wagstaff's (1986) findingthat an increase in proxies for initial assets and lifetime earnings raise health. Noneof the studies has taken my suggestion to treat uncertainty in terms of a probabilitydistribution of depreciation rates in a given period. This could be done by writing thestock of health in period t as

Ht = Ht1 - Et-lHt-1 + It-i + Rt- 1, (57)

where t-1 is the mean depreciation rate and Rt-_ = (t-l - 3t-_)Ht- 1. I leave it tothe reader to explore the implications of this formulation.

7. Health and schooling

An extensive review of the literature conducted by Grossman and Kaestner (1997) sug-gests that years of formal schooling completed is the most important correlate of good

49 Garber and Phelps (1997), Meltzer (1997), and Picone et al. (1998) also introduce uncertainty into a pureconsumption model of the demand for health. I do not discuss the first two studies because they focus oncost-effectiveness analysis. I do not discuss the last one because it emphasizes the behavior of individuals intheir retirement years.

non-increasing absolute risk aversion with regard to that good). In the second model thesame effect is more difficult to sign, although it is positive if an increase in R leads to areduction in M and an increase in the composite good and if the utility function exhibitsnon-increasing relative risk aversion with regard to the composite good.49

Liljas (1998) considers uncertainty in the context of a multiperiod mixed investment-consumption model. Uncertainty takes the form of a random variable that affects thestock of health in period t in an additive fashion. He shows that the stock of health islarger in the stochastic case than in the certainty case. Presumably, this result pertainsto the expected stock of health in period t. The actual stock should be smaller than thestock with certainty given a negative shock. Social insurance that funds part of the lossin income due to illness lowers the optimal stock. Private insurance that also funds partof this loss will not necessarily lower the stock further and may actually increase it ifthe cost of this insurance falls as the stock rises.

To summarize, compared to a model with perfect certainty, the expected value of thestock of health is larger and the optimal quantities of gross investment and health inputsalso are larger in a model with uncertainty. In a pure investment model an increase ininitial assets can cause health and medical care to change, but the direction of theseeffects is ambiguous. Under reasonable assumptions, an increase in the variance of riskraises optimal medical care in a pure consumption model.

How valuable are these results? With the exception of the ambiguity of the asseteffect, they are not very surprising. The variance of risk is extremely difficult to measure.I am not aware of empirical studies that have attempted to include this variable in ademand for health framework. The possibility that the asset effect can be nonzero in apure investment model provides an alternative explanation of Wagstaff's (1986) findingthat an increase in proxies for initial assets and lifetime earnings raise health. Noneof the studies has taken my suggestion to treat uncertainty in terms of a probabilitydistribution of depreciation rates in a given period. This could be done by writing thestock of health in period t as

Ht = Ht1 - Et-lHt-1 + It-i + Rt- 1, (57)

where t-1 is the mean depreciation rate and Rt-_ = (t-l - 3t-_)Ht- 1. I leave it tothe reader to explore the implications of this formulation.

7. Health and schooling

An extensive review of the literature conducted by Grossman and Kaestner (1997) sug-gests that years of formal schooling completed is the most important correlate of good

49 Garber and Phelps (1997), Meltzer (1997), and Picone et al. (1998) also introduce uncertainty into a pureconsumption model of the demand for health. I do not discuss the first two studies because they focus oncost-effectiveness analysis. I do not discuss the last one because it emphasizes the behavior of individuals intheir retirement years.

M. Grossman

health. This finding emerges whether health levels are measured by mortality rates, mor-bidity rates, self-evaluation of health status, or physiological indicators of health, andwhether the units of observation are individuals or groups. The studies also suggest thatschooling is a more important correlate of health than occupation or income, the twoother components of socioeconomic status. This is particularly true when one controlsfor reverse causality from poor health to low income. Of course, schooling is a causaldeterminant of occupation and income, so that the gross effect of schooling on healthmay reflect in part its impact on socioeconomic status. The studies reviewed, however,indicate that a significant portion of the gross schooling effect cannot be traced to therelationship between schooling and income or occupation.

In a broad sense, the observed positive correlation between health and schooling maybe explained in one of three ways. The first argues that there is a causal relationshipthat runs from increases in schooling to increases in health. The second holds that thedirection of causality runs from better health to more schooling. The third argues thatno causal relationship is implied by the correlation; instead, differences in one or more"third variables," such as physical and mental ability and parental characteristics, affectboth health and schooling in the same direction.

It should be noted that these three explanations are not mutually exclusive and can beused to rationalize an observed correlation between any two variables. But from a publicpolicy perspective, it is important to distinguish among them and to obtain quantitativeestimates of their relative magnitudes. Suppose that a stated goal of public policy isto improve the level of health of the population or of certain groups in the population.Given this goal and given the high correlation between health and schooling, it mightappear that one method of implementation would be to increase government outlays onschooling. In fact, Auster et al. (1969) suggest that the rate of return on increases inhealth via higher schooling outlays far exceeds the rate of return on increases in healthvia higher medical care outlays. This argument assumes that the correlation betweenhealth and schooling reflects only the effect of schooling on health. If, however, thecausal relationship was the reverse, or if the third-variable hypothesis was relevant, thenincreased outlays on schooling would not accomplish the goal of improved health.

Causality from schooling to health results when more educated persons are moreefficient producers of health. This efficiency effect can take two forms. Productive effi-ciency pertains to a situation in which the more educated obtain a larger health outputfrom given amounts of endogenous (choice) inputs. This is the effect that I have empha-sized throughout this paper. Allocative efficiency, discussed in detail by Kenkel (2000),pertains to a situation in which schooling increases information about the true effects ofthe inputs on health. For example, the more educated may have more knowledge aboutthe harmful effects of cigarette smoking or about what constitutes an appropriate diet.Allocative efficiency will improve health to the extent that it leads to the selection of abetter input mix.

Causality from schooling to health also results when education changes tastes or pref-erences in a manner that favors health relative to certain other commodities. In somecases the taste hypothesis cannot be distinguished from allocative hypothesis, partic-

M. Grossman

health. This finding emerges whether health levels are measured by mortality rates, mor-bidity rates, self-evaluation of health status, or physiological indicators of health, andwhether the units of observation are individuals or groups. The studies also suggest thatschooling is a more important correlate of health than occupation or income, the twoother components of socioeconomic status. This is particularly true when one controlsfor reverse causality from poor health to low income. Of course, schooling is a causaldeterminant of occupation and income, so that the gross effect of schooling on healthmay reflect in part its impact on socioeconomic status. The studies reviewed, however,indicate that a significant portion of the gross schooling effect cannot be traced to therelationship between schooling and income or occupation.

In a broad sense, the observed positive correlation between health and schooling maybe explained in one of three ways. The first argues that there is a causal relationshipthat runs from increases in schooling to increases in health. The second holds that thedirection of causality runs from better health to more schooling. The third argues thatno causal relationship is implied by the correlation; instead, differences in one or more"third variables," such as physical and mental ability and parental characteristics, affectboth health and schooling in the same direction.

It should be noted that these three explanations are not mutually exclusive and can beused to rationalize an observed correlation between any two variables. But from a publicpolicy perspective, it is important to distinguish among them and to obtain quantitativeestimates of their relative magnitudes. Suppose that a stated goal of public policy isto improve the level of health of the population or of certain groups in the population.Given this goal and given the high correlation between health and schooling, it mightappear that one method of implementation would be to increase government outlays onschooling. In fact, Auster et al. (1969) suggest that the rate of return on increases inhealth via higher schooling outlays far exceeds the rate of return on increases in healthvia higher medical care outlays. This argument assumes that the correlation betweenhealth and schooling reflects only the effect of schooling on health. If, however, thecausal relationship was the reverse, or if the third-variable hypothesis was relevant, thenincreased outlays on schooling would not accomplish the goal of improved health.

Causality from schooling to health results when more educated persons are moreefficient producers of health. This efficiency effect can take two forms. Productive effi-ciency pertains to a situation in which the more educated obtain a larger health outputfrom given amounts of endogenous (choice) inputs. This is the effect that I have empha-sized throughout this paper. Allocative efficiency, discussed in detail by Kenkel (2000),pertains to a situation in which schooling increases information about the true effects ofthe inputs on health. For example, the more educated may have more knowledge aboutthe harmful effects of cigarette smoking or about what constitutes an appropriate diet.Allocative efficiency will improve health to the extent that it leads to the selection of abetter input mix.

Causality from schooling to health also results when education changes tastes or pref-erences in a manner that favors health relative to certain other commodities. In somecases the taste hypothesis cannot be distinguished from allocative hypothesis, partic-

ularly when knowledge of health effects has been available for some time. But in asituation in which the new information becomes available, the allocative efficiency hy-pothesis predicts a more rapid response by the more educated.

Alternatively, the direction of causality may run from better health to more schoolingbecause healthier students may be more efficient producers of additions to the stockof knowledge (or human capital) via formal schooling. Furthermore, this causal pathmay have long lasting effects if past health is an input into current health status. Thus,even for non-students, a positive relationship between health and schooling may reflectreverse causality in the absence of controls for past health.

The "third-variable" explanation is particularly relevant if one thinks that a large un-explained variation in health remains after controlling for schooling and other determi-nants. Studies summarized by Grossman and Kaestner (1997) and results in the relatedfield of investment in human capital and the determinants of earnings [for example,Mincer (1974)], indicate that the percentage of the variation in health explained byschooling is much smaller than the percentage of the variation in earnings explained byschooling. Yet it also is intuitive that health and illness have larger random componentsthan earnings. The third-variable explanation is relevant only if the unaccounted factorswhich affect health are correlated with schooling. Note that both the reverse causalityexplanation and the third-variable explanation indicate that the observed relationshipbetween current health and schooling reflects an omitted variable. In the case of reversecausality, the omitted variable is identified as past or endowed health. In econometricterminology, both explanations fall under the general rubric of biases due to unobservedheterogeneity among individuals.

Kaestner and I [Grossman and Kaestner (1997)] conclude from our extensive reviewof the literature that schooling does in fact have a causal impact on good health. Indrawing this conclusion, we are sensitive to the difficulties of establishing causality inthe social sciences where natural experiments rarely can be performed. Our affirmativeanswer is based on the numerous studies in the US and developing countries that wehave summarized. These studies employ a variety of adult, child, and infant health mea-sures, many different estimation techniques, and controls for a host of third variables.

I leave it up to the reader to evaluate this conclusion after reading the Grossman-Kaestner paper and the studies therein. I also urge the reader to consult my study dealingwith the correlation between health and schooling [Grossman (1975)] because I sketchout a framework in which there are complementary relationships between schoolingand health - the principal components of the stock of human capital - at various stagesin the life cycle. The empirical evidence that Kaestner and I report on causality fromschooling to health as well as on causality from health to schooling underscores thepotential payoffs to the formal development of a model in which the stocks of healthand knowledge are determined simultaneously.

In the remainder of this section, I want to address one challenge of the conclusionthat the role of schooling is causal: the time preference hypothesis first proposed byFuchs (1982). Fuchs argues that persons who are more future oriented (who have a highdegree of time preference for the future) attend school for longer periods of time and

ularly when knowledge of health effects has been available for some time. But in asituation in which the new information becomes available, the allocative efficiency hy-pothesis predicts a more rapid response by the more educated.

Alternatively, the direction of causality may run from better health to more schoolingbecause healthier students may be more efficient producers of additions to the stockof knowledge (or human capital) via formal schooling. Furthermore, this causal pathmay have long lasting effects if past health is an input into current health status. Thus,even for non-students, a positive relationship between health and schooling may reflectreverse causality in the absence of controls for past health.

The "third-variable" explanation is particularly relevant if one thinks that a large un-explained variation in health remains after controlling for schooling and other determi-nants. Studies summarized by Grossman and Kaestner (1997) and results in the relatedfield of investment in human capital and the determinants of earnings [for example,Mincer (1974)], indicate that the percentage of the variation in health explained byschooling is much smaller than the percentage of the variation in earnings explained byschooling. Yet it also is intuitive that health and illness have larger random componentsthan earnings. The third-variable explanation is relevant only if the unaccounted factorswhich affect health are correlated with schooling. Note that both the reverse causalityexplanation and the third-variable explanation indicate that the observed relationshipbetween current health and schooling reflects an omitted variable. In the case of reversecausality, the omitted variable is identified as past or endowed health. In econometricterminology, both explanations fall under the general rubric of biases due to unobservedheterogeneity among individuals.

Kaestner and I [Grossman and Kaestner (1997)] conclude from our extensive reviewof the literature that schooling does in fact have a causal impact on good health. Indrawing this conclusion, we are sensitive to the difficulties of establishing causality inthe social sciences where natural experiments rarely can be performed. Our affirmativeanswer is based on the numerous studies in the US and developing countries that wehave summarized. These studies employ a variety of adult, child, and infant health mea-sures, many different estimation techniques, and controls for a host of third variables.

I leave it up to the reader to evaluate this conclusion after reading the Grossman-Kaestner paper and the studies therein. I also urge the reader to consult my study dealingwith the correlation between health and schooling [Grossman (1975)] because I sketchout a framework in which there are complementary relationships between schoolingand health - the principal components of the stock of human capital - at various stagesin the life cycle. The empirical evidence that Kaestner and I report on causality fromschooling to health as well as on causality from health to schooling underscores thepotential payoffs to the formal development of a model in which the stocks of healthand knowledge are determined simultaneously.

In the remainder of this section, I want to address one challenge of the conclusionthat the role of schooling is causal: the time preference hypothesis first proposed byFuchs (1982). Fuchs argues that persons who are more future oriented (who have a highdegree of time preference for the future) attend school for longer periods of time and

make larger investments in health. Thus, the effect of schooling on health is biased ifone fails to control for time preference.

The time preference hypothesis is analogous to the hypothesis that the positive effectof schooling on earnings is biased upward by the omission of ability. In each case a well-established relationship between schooling and an outcome (earnings or health) is chal-lenged because a hard-to-measure variable (ability or time preference) has been omitted.Much ink has been spilled on this issue in the human capital literature. Attempts to in-clude proxies for ability in earnings functions have resulted in very modest reductionsin the schooling coefficient [for example, Griliches and Mason (1972), Hause (1972)].Proponents of the ability hypothesis have attributed the modest reductions to measure-ment error in these proxies [for example, Goldberger (1974)]. More recent efforts havesought instruments that are correlated with schooling but not correlated with ability [forexample, Angrist and Krueger (1991)]. These efforts have produced the somewhat sur-prising finding that the schooling coefficient increases when the instrumental variablesprocedure is employed. A cynic might conclude that the way to destroy any empiricalregularity is to attribute it to an unmeasured variable, especially if the theory with regardto the relevance of this variable is not well developed. 50

Nevertheless, the time preference hypothesis is important because it is related to re-cent and potentially very rich theoretical models in which preferences are endogenous[Becker and Murphy (1988), Becker (1996), Becker and Mulligan (1997)]. Differencesin time preference among individuals will not generate differences in investments in hu-man capital unless certain other conditions are met. One condition is that the ability tofinance these investments by borrowing is limited, so that they must be funded to someextent by foregoing current consumption. Even if the capital market is perfect, the re-turns on an investment in schooling depend on hours of work if schooling raises marketproductivity by a larger percentage than it raises nonmarket productivity. Individualswho are more future oriented desire relatively more leisure at older ages. Therefore,they work more at younger ages and have a higher discounted marginal benefit on agiven investment than persons who are more present oriented. If health enters the utilityfunction, persons who discount the future less heavily will have higher health levelsduring most stages of the life cycle. Hence, a positive relationship between schoolingand health does not necessarily imply causality.

Since the conditions that generate causal effects of time preference on schooling andhealth are plausible, attempts to control for time preference in estimating the schoolingcoefficient in a health outcome equation are valuable. Fuchs (1982) measures time pref-erence in a telephone survey by asking respondents questions in which they chose be-tween a sum of money now and a larger sum in the future. He includes an index of timepreference in a multiple regression in which health status is the dependent variable and

50 See Grossman and Kaestner (1997) for a model in which ability should be omitted from the reducedform earnings function even though it enters the structural production function and has a causal impact onschooling.

398 M. Grossmanr

make larger investments in health. Thus, the effect of schooling on health is biased ifone fails to control for time preference.

The time preference hypothesis is analogous to the hypothesis that the positive effectof schooling on earnings is biased upward by the omission of ability. In each case a well-established relationship between schooling and an outcome (earnings or health) is chal-lenged because a hard-to-measure variable (ability or time preference) has been omitted.Much ink has been spilled on this issue in the human capital literature. Attempts to in-clude proxies for ability in earnings functions have resulted in very modest reductionsin the schooling coefficient [for example, Griliches and Mason (1972), Hause (1972)].Proponents of the ability hypothesis have attributed the modest reductions to measure-ment error in these proxies [for example, Goldberger (1974)]. More recent efforts havesought instruments that are correlated with schooling but not correlated with ability [forexample, Angrist and Krueger (1991)]. These efforts have produced the somewhat sur-prising finding that the schooling coefficient increases when the instrumental variablesprocedure is employed. A cynic might conclude that the way to destroy any empiricalregularity is to attribute it to an unmeasured variable, especially if the theory with regardto the relevance of this variable is not well developed. 50

Nevertheless, the time preference hypothesis is important because it is related to re-cent and potentially very rich theoretical models in which preferences are endogenous[Becker and Murphy (1988), Becker (1996), Becker and Mulligan (1997)]. Differencesin time preference among individuals will not generate differences in investments in hu-man capital unless certain other conditions are met. One condition is that the ability tofinance these investments by borrowing is limited, so that they must be funded to someextent by foregoing current consumption. Even if the capital market is perfect, the re-turns on an investment in schooling depend on hours of work if schooling raises marketproductivity by a larger percentage than it raises nonmarket productivity. Individualswho are more future oriented desire relatively more leisure at older ages. Therefore,they work more at younger ages and have a higher discounted marginal benefit on agiven investment than persons who are more present oriented. If health enters the utilityfunction, persons who discount the future less heavily will have higher health levelsduring most stages of the life cycle. Hence, a positive relationship between schoolingand health does not necessarily imply causality.

Since the conditions that generate causal effects of time preference on schooling andhealth are plausible, attempts to control for time preference in estimating the schoolingcoefficient in a health outcome equation are valuable. Fuchs (1982) measures time pref-erence in a telephone survey by asking respondents questions in which they chose be-tween a sum of money now and a larger sum in the future. He includes an index of timepreference in a multiple regression in which health status is the dependent variable and

50 See Grossman and Kaestner (1997) for a model in which ability should be omitted from the reducedform earnings function even though it enters the structural production function and has a causal impact onschooling.

398 M. Grossmanr

schooling is one of the independent variables. Fuchs is not able to demonstrate that theschooling effect is due to time preference. The latter variable has a negative regressioncoefficient, but it is not statistically significant. When time preference and schooling areentered simultaneously, the latter dominates the former. These results must be regardedas preliminary because they are based on one small sample of adults on Long Island andon exploratory measures of time preference.

Farrell and Fuchs (1982) explore the time preference hypothesis in the context ofcigarette smoking using interviews conducted in 1979 by the Stanford Heart DiseasePrevention Program in four small agricultural cities in California. They examine thesmoking behavior of white non-Hispanics who were not students at the time of thesurvey, had completed 12 to 18 years of schooling, and were at least 24 years old. Thepresence of retrospective information on cigarette smoking at ages 17 and 24 allowsthem to relate smoking at these two ages to years of formal schooling completed by 1979for cohorts who reached age 17 before and after the widespread diffusion of informationconcerning the harmful effects of cigarette smoking on health.

Farrell and Fuchs find that the negative relationship between schooling and smoking,which rises in absolute value for cohorts born after 1953, does not increase between theages of 17 and 24. Since the individuals were all in the same school grade at age 17, theadditional schooling obtained between that age and age 24 cannot be the cause of dif-ferential smoking behavior at age 24, according to the authors. Based on these results,Farrell and Fuchs reject the hypothesis that schooling is a causal factor in smoking be-havior in favor of the view that a third variable causes both. Since the strong negativerelationship between schooling and smoking developed only after the spread of infor-mation concerning the harmful effects of smoking, they argue that the same mechanismmay generate the schooling-health relationship.

A different interpretation of the Farrell and Fuchs finding emerges if one assumes thatconsumers are farsighted. The current consumption of cigarettes leads to more illnessand less time for work in the future. The cost of this lost time is higher for persons withhigher wage rates who have made larger investments in human capital. Thus, the costsof smoking in high school are greater for persons who plan to make larger investmentsin human capital.

Berger and Leigh (1989) have developed an extremely useful methodology for dis-entangling the schooling effect from the time preference effect. Their methodologyamounts to treating schooling as an endogenous variable in the health equation andestimating the equation by a variant of two-stage least squares. If the instrumental vari-ables used to predict schooling in the first stage are uncorrelated with time preference,this technique yields an unbiased estimate of the schooling coefficient. Since the frame-work generates a recursive model with correlated errors, exogenous variables that areunique to the health equation are not used to predict schooling.

Berger and Leigh apply their methodology to two data sets: the first National Healthand Nutrition Examination Survey (NHANES I) and the National Longitudinal Surveyof Young Men (NLS). In NHANES I, health is measured by blood pressure, and separateequations are obtained for persons aged 20 through 40 and over age 40 in the period

schooling is one of the independent variables. Fuchs is not able to demonstrate that theschooling effect is due to time preference. The latter variable has a negative regressioncoefficient, but it is not statistically significant. When time preference and schooling areentered simultaneously, the latter dominates the former. These results must be regardedas preliminary because they are based on one small sample of adults on Long Island andon exploratory measures of time preference.

Farrell and Fuchs (1982) explore the time preference hypothesis in the context ofcigarette smoking using interviews conducted in 1979 by the Stanford Heart DiseasePrevention Program in four small agricultural cities in California. They examine thesmoking behavior of white non-Hispanics who were not students at the time of thesurvey, had completed 12 to 18 years of schooling, and were at least 24 years old. Thepresence of retrospective information on cigarette smoking at ages 17 and 24 allowsthem to relate smoking at these two ages to years of formal schooling completed by 1979for cohorts who reached age 17 before and after the widespread diffusion of informationconcerning the harmful effects of cigarette smoking on health.

Farrell and Fuchs find that the negative relationship between schooling and smoking,which rises in absolute value for cohorts born after 1953, does not increase between theages of 17 and 24. Since the individuals were all in the same school grade at age 17, theadditional schooling obtained between that age and age 24 cannot be the cause of dif-ferential smoking behavior at age 24, according to the authors. Based on these results,Farrell and Fuchs reject the hypothesis that schooling is a causal factor in smoking be-havior in favor of the view that a third variable causes both. Since the strong negativerelationship between schooling and smoking developed only after the spread of infor-mation concerning the harmful effects of smoking, they argue that the same mechanismmay generate the schooling-health relationship.

A different interpretation of the Farrell and Fuchs finding emerges if one assumes thatconsumers are farsighted. The current consumption of cigarettes leads to more illnessand less time for work in the future. The cost of this lost time is higher for persons withhigher wage rates who have made larger investments in human capital. Thus, the costsof smoking in high school are greater for persons who plan to make larger investmentsin human capital.

Berger and Leigh (1989) have developed an extremely useful methodology for dis-entangling the schooling effect from the time preference effect. Their methodologyamounts to treating schooling as an endogenous variable in the health equation andestimating the equation by a variant of two-stage least squares. If the instrumental vari-ables used to predict schooling in the first stage are uncorrelated with time preference,this technique yields an unbiased estimate of the schooling coefficient. Since the frame-work generates a recursive model with correlated errors, exogenous variables that areunique to the health equation are not used to predict schooling.

Berger and Leigh apply their methodology to two data sets: the first National Healthand Nutrition Examination Survey (NHANES I) and the National Longitudinal Surveyof Young Men (NLS). In NHANES I, health is measured by blood pressure, and separateequations are obtained for persons aged 20 through 40 and over age 40 in the period

M. Grossman

1971 through 1975. The schooling equation is identified by ancestry and by averagereal per capita income and average real per capita expenditures on education in the statein which an individual resided from the year of birth to age 6. These variables enterthe schooling equation but are excluded from the health equation. In the NLS, health ismeasured by a dichotomous variable that identifies men who in 1976 reported that healthlimited or prevented them from working and alternatively by a dichotomous variablethat identifies the presence of a functional health limitation. The men in the samplewere between the ages of 24 and 34 in 1976, had left school by that year, and reportedno health limitations in 1966 (the first year of the survey). The schooling equation isidentified by IQ, Knowledge of Work test scores, and parents' schooling.

Results from the NLS show that the schooling coefficient rises in absolute value whenpredicted schooling replaces actual schooling, and when health is measured by worklimitation. When health is measured by functional limitation, the two-stage least squaresschooling coefficient is approximately equal to the ordinary least squares coefficient, al-though the latter is estimated with more precision. For persons aged 20 through 40 inNHANES I, schooling has a larger impact on blood pressure in absolute value in thetwo-stage regressions. For persons over age 40, however, the predicted value of school.-ing has a positive and insignificant regression coefficient. Except for the last finding,these results are inconsistent with the time preference hypothesis and consistent withthe hypothesis that schooling causes health.

In another application of the same methodology, Leigh and Dhir (1997) focus onthe relationship between schooling and health among persons ages 65 and over in the1986 wave of the Panel Survey of Income Dynamics (PSID). Health is measured by adisability index comprised of answers to six activities of daily living and by a measure ofexercise frequency. Responses to questions asked in 1972 concerning the ability to delaygratification are used to form an index of time preference. Instruments for schoolinginclude parents' schooling, parents' income, and state of residence in childhood. Theschooling variable is associated with better health and more exercise whether it is treatedas exogenous or endogenous.

Sander (1995a, 1995b) has applied the methodology developed by Berger and Leighto the relationship between schooling and cigarette smoking studied by Farrell andFuchs (1982). His data consist of the 1986-1991 waves of the National Opinion Re-search Center's General Social Survey. In the first paper the outcome is the probabilityof quitting smoking, while in the second the outcome is the probability of smoking. Sep-arate probit equations are obtained for men and women ages 25 and older. Instrumentsfor schooling include father's schooling, mother's schooling, rural residence at age 16,region of residence at age 16, and number of siblings.

In general schooling has a negative effect on smoking participation and a positive ef-fect on the probability of quitting smoking. These results are not sensitive to the use ofpredicted as opposed to actual schooling in the probit regressions. Moreover, the appli-cation of the Wu-Hausman endogeneity test [Wu (1973), Hausman (1978)] in the quitequation suggest that schooling is exogenous in this equation. Thus, Sander's results,

M. Grossman

1971 through 1975. The schooling equation is identified by ancestry and by averagereal per capita income and average real per capita expenditures on education in the statein which an individual resided from the year of birth to age 6. These variables enterthe schooling equation but are excluded from the health equation. In the NLS, health ismeasured by a dichotomous variable that identifies men who in 1976 reported that healthlimited or prevented them from working and alternatively by a dichotomous variablethat identifies the presence of a functional health limitation. The men in the samplewere between the ages of 24 and 34 in 1976, had left school by that year, and reportedno health limitations in 1966 (the first year of the survey). The schooling equation isidentified by IQ, Knowledge of Work test scores, and parents' schooling.

Results from the NLS show that the schooling coefficient rises in absolute value whenpredicted schooling replaces actual schooling, and when health is measured by worklimitation. When health is measured by functional limitation, the two-stage least squaresschooling coefficient is approximately equal to the ordinary least squares coefficient, al-though the latter is estimated with more precision. For persons aged 20 through 40 inNHANES I, schooling has a larger impact on blood pressure in absolute value in thetwo-stage regressions. For persons over age 40, however, the predicted value of school.-ing has a positive and insignificant regression coefficient. Except for the last finding,these results are inconsistent with the time preference hypothesis and consistent withthe hypothesis that schooling causes health.

In another application of the same methodology, Leigh and Dhir (1997) focus onthe relationship between schooling and health among persons ages 65 and over in the1986 wave of the Panel Survey of Income Dynamics (PSID). Health is measured by adisability index comprised of answers to six activities of daily living and by a measure ofexercise frequency. Responses to questions asked in 1972 concerning the ability to delaygratification are used to form an index of time preference. Instruments for schoolinginclude parents' schooling, parents' income, and state of residence in childhood. Theschooling variable is associated with better health and more exercise whether it is treatedas exogenous or endogenous.

Sander (1995a, 1995b) has applied the methodology developed by Berger and Leighto the relationship between schooling and cigarette smoking studied by Farrell andFuchs (1982). His data consist of the 1986-1991 waves of the National Opinion Re-search Center's General Social Survey. In the first paper the outcome is the probabilityof quitting smoking, while in the second the outcome is the probability of smoking. Sep-arate probit equations are obtained for men and women ages 25 and older. Instrumentsfor schooling include father's schooling, mother's schooling, rural residence at age 16,region of residence at age 16, and number of siblings.

In general schooling has a negative effect on smoking participation and a positive ef-fect on the probability of quitting smoking. These results are not sensitive to the use ofpredicted as opposed to actual schooling in the probit regressions. Moreover, the appli-cation of the Wu-Hausman endogeneity test [Wu (1973), Hausman (1978)] in the quitequation suggest that schooling is exogenous in this equation. Thus, Sander's results,

like Berger and Leigh's and Leigh and Dhir's results, are inconsistent with the timepreference hypothesis.

The aforementioned conclusion rests on the assumption that the instruments used topredict schooling in the first stage are uncorrelated with time preference. The validityof this assumption is most plausible in the case of measures such as real per capitaincome and real per capita outlays on education in the state in which an individualresided from birth to age 6 (used by Berger and Leigh in NHANES I), state of residencein childhood (used by Leigh and Dhir in the PSID) and rural residence at age 16 andregion of residence at that age (used by Sander). The validity of the assumption is lessplausible in the case of measures such as parents' schooling (used by Sander and byBerger and Leigh in the NLS and by Leigh and Dhir in the PSID) and parents' income(used by Leigh and Dhir in the PSID).

Given this and the inherent difficulty in Fuchs's (1982) and Leigh and Dhir's(1997) attempts to measure time preference directly, definitive evidence with re-gard to the time preference hypothesis still is lacking. Moreover, Sander (1995a,1995b) presents national data showing a much larger downward trend in the proba-bility of smoking and a much larger upward trend in the probability of quitting smokingbetween 1966 and 1987 as the level of education rises. Since information concerningthe harmful effects of smoking was widespread by the early 1980s, these results are notconsistent with an allocative efficiency argument that the more educated are better ableto process new information.

Becker and Murphy's (1988) theoretical model of rational addiction predicts thatpersons who discount the future heavily are more likely to participate in such addic-tive behaviors as cigarette smoking. Becker et al. (1991) show that the higher educatedpeople are more responsive to changes in the harmful future consequences of the con-sumption of addictive goods because they are more future oriented. Thus, the trends justcited are consistent with a negative relationship between schooling and the rate of timepreference for the present.

Proponents of the time preference hypothesis assume that a reduction in the rate oftime preference for the present causes years of formal schooling to rise. On the otherhand, Becker and Mulligan (1997) argue that causality may run in the opposite direc-tion: namely, an increase in schooling may cause the rate of time preference for thepresent to fall (may cause the rate of time preference for the future to rise). In mostmodels of optimal consumption over the life cycle, consumers maximize a lifetime util-ity function defined as the discounted sum or present value of utility in each period orat each age. The discount factor () is given by PB = 1/(1 + g), where g is the rate oftime preference for the present. Becker and Mulligan point out that the present value ofutility is higher the smaller is the rate of time preference for the present. Hence, con-sumers have incentives to make investments that lower the rate of time preference forthe present.

Becker and Mulligan then show that the marginal costs of investments that lower timepreference fall and the marginal benefits rise as income or wealth rises. Marginal bene-fits also are greater when the length of life is greater. Hence, the equilibrium rate of time

like Berger and Leigh's and Leigh and Dhir's results, are inconsistent with the timepreference hypothesis.

The aforementioned conclusion rests on the assumption that the instruments used topredict schooling in the first stage are uncorrelated with time preference. The validityof this assumption is most plausible in the case of measures such as real per capitaincome and real per capita outlays on education in the state in which an individualresided from birth to age 6 (used by Berger and Leigh in NHANES I), state of residencein childhood (used by Leigh and Dhir in the PSID) and rural residence at age 16 andregion of residence at that age (used by Sander). The validity of the assumption is lessplausible in the case of measures such as parents' schooling (used by Sander and byBerger and Leigh in the NLS and by Leigh and Dhir in the PSID) and parents' income(used by Leigh and Dhir in the PSID).

Given this and the inherent difficulty in Fuchs's (1982) and Leigh and Dhir's(1997) attempts to measure time preference directly, definitive evidence with re-gard to the time preference hypothesis still is lacking. Moreover, Sander (1995a,1995b) presents national data showing a much larger downward trend in the proba-bility of smoking and a much larger upward trend in the probability of quitting smokingbetween 1966 and 1987 as the level of education rises. Since information concerningthe harmful effects of smoking was widespread by the early 1980s, these results are notconsistent with an allocative efficiency argument that the more educated are better ableto process new information.

Becker and Murphy's (1988) theoretical model of rational addiction predicts thatpersons who discount the future heavily are more likely to participate in such addic-tive behaviors as cigarette smoking. Becker et al. (1991) show that the higher educatedpeople are more responsive to changes in the harmful future consequences of the con-sumption of addictive goods because they are more future oriented. Thus, the trends justcited are consistent with a negative relationship between schooling and the rate of timepreference for the present.

Proponents of the time preference hypothesis assume that a reduction in the rate oftime preference for the present causes years of formal schooling to rise. On the otherhand, Becker and Mulligan (1997) argue that causality may run in the opposite direc-tion: namely, an increase in schooling may cause the rate of time preference for thepresent to fall (may cause the rate of time preference for the future to rise). In mostmodels of optimal consumption over the life cycle, consumers maximize a lifetime util-ity function defined as the discounted sum or present value of utility in each period orat each age. The discount factor () is given by PB = 1/(1 + g), where g is the rate oftime preference for the present. Becker and Mulligan point out that the present value ofutility is higher the smaller is the rate of time preference for the present. Hence, con-sumers have incentives to make investments that lower the rate of time preference forthe present.

Becker and Mulligan then show that the marginal costs of investments that lower timepreference fall and the marginal benefits rise as income or wealth rises. Marginal bene-fits also are greater when the length of life is greater. Hence, the equilibrium rate of time

M. Grossman

preference falls as the level of education rises because education raises income and lifeexpectancy. Moreover, the more educated may be more efficient in making investmentsthat lower the rate of time preference for the present - a form of productive efficiencynot associated with health production. To quote Becker and Mulligan: "Schooling alsodetermines ... [investments in time preference] partly through the study of history andother subjects, for schooling focuses students' attention on the future. Schooling cancommunicate images of the situations and difficulties of adult life, which are the fu-ture of childhood and adolescence. In addition, through repeated practice at problemsolving, schooling helps children learn the art of scenario simulation. Thus, educatedpeople should be more productive at reducing the remoteness of future pleasures" (pp.735-736).

Becker and Mulligan's argument amounts to a third causal mechanism in addition toproductive and allocative efficiency in health production via which schooling can causehealth. Econometrically, the difference between their model and Fuchs's model can bespecified as follows:

H = alY+ a2E+a3g, (58)

g = a4 Y +asE, (59)

E = a6g, (60)

Y = oa7 E. (61)

In this system H is health, Y is permanent income, E is years of formal schooling com-pleted, g is time preference for the present, and the disturbance terms are suppressed.The first equation is a demand for health function in which the coefficient of E reflectsproductive or allocative efficiency or both. Fuchs assumes that as5 is zero. Hence, thecoefficient of E in the first equation is biased if g is omitted.

In one version of their model, Becker and Mulligan assume that a6 is zero, althoughin a more general formulation they allow this coefficient to be nonzero. Given that oz6 iszero, and substituting the second equation into the first, one obtains

H = (al + a4Ca3)Y + (2 + a53)E. (62)

The coefficient of Y in the last equation reflects both the direct effect of income onhealth (al) and the indirect effect of income on health through time preference (43).Similarly, the coefficient of E reflects both the direct efficiency effect (2) and theindirect effect of schooling on health through time preference (5cr3)

Suppose that the direct efficiency effect of schooling (2) is zero. In Fuchs's model,if health is regressed on income and schooling as represented by solving Equation (60)for g, the expected value of the schooling coefficient is Co3/or6. This coefficient re-flects causality from time preference to schooling. In Becker and Mulligan's model theschooling coefficient is a3a5. This coefficient reflects causality from schooling to timepreference. The equation that expresses income as a function of schooling stresses that

M. Grossman

preference falls as the level of education rises because education raises income and lifeexpectancy. Moreover, the more educated may be more efficient in making investmentsthat lower the rate of time preference for the present - a form of productive efficiencynot associated with health production. To quote Becker and Mulligan: "Schooling alsodetermines ... [investments in time preference] partly through the study of history andother subjects, for schooling focuses students' attention on the future. Schooling cancommunicate images of the situations and difficulties of adult life, which are the fu-ture of childhood and adolescence. In addition, through repeated practice at problemsolving, schooling helps children learn the art of scenario simulation. Thus, educatedpeople should be more productive at reducing the remoteness of future pleasures" (pp.735-736).

Becker and Mulligan's argument amounts to a third causal mechanism in addition toproductive and allocative efficiency in health production via which schooling can causehealth. Econometrically, the difference between their model and Fuchs's model can bespecified as follows:

H = alY+ a2E+a3g, (58)

g = a4 Y +asE, (59)

E = a6g, (60)

Y = oa7 E. (61)

In this system H is health, Y is permanent income, E is years of formal schooling com-pleted, g is time preference for the present, and the disturbance terms are suppressed.The first equation is a demand for health function in which the coefficient of E reflectsproductive or allocative efficiency or both. Fuchs assumes that as5 is zero. Hence, thecoefficient of E in the first equation is biased if g is omitted.

In one version of their model, Becker and Mulligan assume that a6 is zero, althoughin a more general formulation they allow this coefficient to be nonzero. Given that oz6 iszero, and substituting the second equation into the first, one obtains

H = (al + a4Ca3)Y + (2 + a53)E. (62)

The coefficient of Y in the last equation reflects both the direct effect of income onhealth (al) and the indirect effect of income on health through time preference (43).Similarly, the coefficient of E reflects both the direct efficiency effect (2) and theindirect effect of schooling on health through time preference (5cr3)

Suppose that the direct efficiency effect of schooling (2) is zero. In Fuchs's model,if health is regressed on income and schooling as represented by solving Equation (60)for g, the expected value of the schooling coefficient is Co3/or6. This coefficient re-flects causality from time preference to schooling. In Becker and Mulligan's model theschooling coefficient is a3a5. This coefficient reflects causality from schooling to timepreference. The equation that expresses income as a function of schooling stresses that

schooling has indirect effects on health via income. Becker and Mulligan would includehealth as a determinant of time preference in the second equation because health lowersmortality, raises future utility levels, and increases incentives to make investments thatlower the rate of time preference.

Becker and Mulligan's model appears to contain useful insights in considering in-tergenerational relationships between parents and children. For example, parents canraise their children's future health, including their adulthood health, by making themmore future oriented. Note that years of formal schooling completed is a time-invariantvariable beyond approximately age 30, while adult health is not time invariant. Thus,parents probably have a more important direct impact on the former than the latter. Bymaking investments that raise their offsprings' schooling, parents also induce them tomake investments that lower their rate of time preference for the present and thereforeraise their adult health.

Becker and Mulligan suggest a more definitive and concrete way to measure timepreference and incorporate it into estimates of health demand functions than thosethat have been attempted to date. They point out that the natural logarithm of the ra-tio of consumption between consecutive time periods (In L) is approximately equal toa [ln(1 + r) - ln(l + g)], where o is the intertemporal elasticity of substitution in con-sumption, r is the market rate of interest, and g is the rate of time preference for thepresent. If oa and r do not vary among individuals, variations in In L capture variationsin time preference. With panel data, In L can be included as a regressor in the health de-mand function Since Becker and Mulligan stress the endogeneity of time preference andits dependence on schooling, simultaneous equations techniques appear to be required.Identification of this model will not be easy, but success in this area has the potential togreatly inform public policy.

To illustrate the last point, suppose that most of the effect of schooling on health oper-ates through time preference. Then school-based programs to promote health knowledgein areas characterized by low levels of income and education may have much smallerpayoffs than programs that encourage the investments in time preference made by themore educated. Indeed, in an ever-changing world in which new information constantlybecomes available, general interventions that encourage future-oriented behavior mayhave much larger rates of return in the long run than specific interventions designed, forexample, to discourage cigarette smoking, alcohol abuse, or the use of illegal drugs.

There appear to be important interactions between Becker and Mulligan's theory ofthe endogenous determination of time preference and Becker and Murphy's (1988) the-ory of rational addiction. Such addictive behaviors as cigarette smoking, excessive alco-hol use, and the consumption of illegal drugs have demonstrated adverse health effects.Increased consumption of these goods raises present utility but lowers future utility. Ac-cording to Becker and Mulligan (1997, p. 744)), "Since a decline in future utility reducesthe benefits from a lower discount on future utilities, greater consumption of harmfulsubstances would lead to higher rates of time preference by discouraging investmentsin lowering these rates ... " This is the converse of Becker and Murphy's result thatpeople who discount the future more heavily are more likely to become addicted. Thus,

schooling has indirect effects on health via income. Becker and Mulligan would includehealth as a determinant of time preference in the second equation because health lowersmortality, raises future utility levels, and increases incentives to make investments thatlower the rate of time preference.

Becker and Mulligan's model appears to contain useful insights in considering in-tergenerational relationships between parents and children. For example, parents canraise their children's future health, including their adulthood health, by making themmore future oriented. Note that years of formal schooling completed is a time-invariantvariable beyond approximately age 30, while adult health is not time invariant. Thus,parents probably have a more important direct impact on the former than the latter. Bymaking investments that raise their offsprings' schooling, parents also induce them tomake investments that lower their rate of time preference for the present and thereforeraise their adult health.

Becker and Mulligan suggest a more definitive and concrete way to measure timepreference and incorporate it into estimates of health demand functions than thosethat have been attempted to date. They point out that the natural logarithm of the ra-tio of consumption between consecutive time periods (In L) is approximately equal toa [ln(1 + r) - ln(l + g)], where o is the intertemporal elasticity of substitution in con-sumption, r is the market rate of interest, and g is the rate of time preference for thepresent. If oa and r do not vary among individuals, variations in In L capture variationsin time preference. With panel data, In L can be included as a regressor in the health de-mand function Since Becker and Mulligan stress the endogeneity of time preference andits dependence on schooling, simultaneous equations techniques appear to be required.Identification of this model will not be easy, but success in this area has the potential togreatly inform public policy.

To illustrate the last point, suppose that most of the effect of schooling on health oper-ates through time preference. Then school-based programs to promote health knowledgein areas characterized by low levels of income and education may have much smallerpayoffs than programs that encourage the investments in time preference made by themore educated. Indeed, in an ever-changing world in which new information constantlybecomes available, general interventions that encourage future-oriented behavior mayhave much larger rates of return in the long run than specific interventions designed, forexample, to discourage cigarette smoking, alcohol abuse, or the use of illegal drugs.

There appear to be important interactions between Becker and Mulligan's theory ofthe endogenous determination of time preference and Becker and Murphy's (1988) the-ory of rational addiction. Such addictive behaviors as cigarette smoking, excessive alco-hol use, and the consumption of illegal drugs have demonstrated adverse health effects.Increased consumption of these goods raises present utility but lowers future utility. Ac-cording to Becker and Mulligan (1997, p. 744)), "Since a decline in future utility reducesthe benefits from a lower discount on future utilities, greater consumption of harmfulsubstances would lead to higher rates of time preference by discouraging investmentsin lowering these rates ... " This is the converse of Becker and Murphy's result thatpeople who discount the future more heavily are more likely to become addicted. Thus,

". .. harmful addictions induce even rational persons to discount the future more heavily,which in turn may lead them to become more addicted" [Becker and Mulligan (1997,p. 744)1.

It is well known that cigarette smoking and excessive alcohol abuse begin early in life[for example, Grossman et al. (1993)]. Moreover, bandwagon or peer effects are muchmore important in the case of youth smoking or alcohol consumption than in the case ofadult smoking or alcohol consumption. The two-way causality between addiction andtime preference and the importance of peer pressure explain why parents who care aboutthe welfare of their children have large incentives to make investments that make theirchildren more future oriented. These forces may also account for the relatively largeimpact of schooling on health with health knowledge held constant reported by Kenkel(1991).

Some parents may ignore or be unaware of the benefits of investments in time prefer-ence. Given society's concern with the welfare of its children, subsidies to school-basedprograms that make children more future oriented may be warranted. But much more re-search dealing with the determinants of time preference and its relationship with school-ing and health is required before these programs can be formulated and implemented ina cost-effective manner.

8. Conclusions

Most of the chapters in this Handbook focus on various aspects of the markets for medi-cal care services and health insurance. This focus is required to understand the determi-nants of prices, quantities, and expenditures in these markets. The main message of mypaper is that a very different theoretical paradigm is required to understand the deter-minants of health outcomes. I have tried to convince the reader that the human capitalmodel of the demand for health provides the framework to conduct investigations ofthese outcomes. The model emphasizes the difference between health as an output andmedical care as one of many inputs into the production of health and the equally im-portant difference between health capital and other forms of human capital. It providesa theoretical framework for making predictions about the impacts of many variables onhealth and an empirical framework for testing these predictions.

Future theoretical efforts will be especially useful if they consider the joint deter-mination of health and schooling and the interactions between these two variables andtime preference for the present. A model in which both the stock of health and the stockof knowledge (schooling) are endogenous does not necessarily generate causality be-tween the two. Individuals, however, typically stop investing in schooling at relativelyyoung ages but rarely stop investing in health. I have a "hunch" that a dynamic modelthat takes account of these patterns will generate effects of an endogenously determinedschooling variable on health in the health demand function if schooling has a causalimpact on productive efficiency or time preference.

Future empirical efforts will be especially useful if they employ longitudinaldatabases with a variety of health outputs, health inputs, and direct and indirect (for

404 M. Grossman

". .. harmful addictions induce even rational persons to discount the future more heavily,which in turn may lead them to become more addicted" [Becker and Mulligan (1997,p. 744)1.

It is well known that cigarette smoking and excessive alcohol abuse begin early in life[for example, Grossman et al. (1993)]. Moreover, bandwagon or peer effects are muchmore important in the case of youth smoking or alcohol consumption than in the case ofadult smoking or alcohol consumption. The two-way causality between addiction andtime preference and the importance of peer pressure explain why parents who care aboutthe welfare of their children have large incentives to make investments that make theirchildren more future oriented. These forces may also account for the relatively largeimpact of schooling on health with health knowledge held constant reported by Kenkel(1991).

Some parents may ignore or be unaware of the benefits of investments in time prefer-ence. Given society's concern with the welfare of its children, subsidies to school-basedprograms that make children more future oriented may be warranted. But much more re-search dealing with the determinants of time preference and its relationship with school-ing and health is required before these programs can be formulated and implemented ina cost-effective manner.

8. Conclusions

Most of the chapters in this Handbook focus on various aspects of the markets for medi-cal care services and health insurance. This focus is required to understand the determi-nants of prices, quantities, and expenditures in these markets. The main message of mypaper is that a very different theoretical paradigm is required to understand the deter-minants of health outcomes. I have tried to convince the reader that the human capitalmodel of the demand for health provides the framework to conduct investigations ofthese outcomes. The model emphasizes the difference between health as an output andmedical care as one of many inputs into the production of health and the equally im-portant difference between health capital and other forms of human capital. It providesa theoretical framework for making predictions about the impacts of many variables onhealth and an empirical framework for testing these predictions.

Future theoretical efforts will be especially useful if they consider the joint deter-mination of health and schooling and the interactions between these two variables andtime preference for the present. A model in which both the stock of health and the stockof knowledge (schooling) are endogenous does not necessarily generate causality be-tween the two. Individuals, however, typically stop investing in schooling at relativelyyoung ages but rarely stop investing in health. I have a "hunch" that a dynamic modelthat takes account of these patterns will generate effects of an endogenously determinedschooling variable on health in the health demand function if schooling has a causalimpact on productive efficiency or time preference.

Future empirical efforts will be especially useful if they employ longitudinaldatabases with a variety of health outputs, health inputs, and direct and indirect (for

404 M. Grossman

example, the rate of growth in total consumption) measures of time preference at threeor more different ages. This is the type of data required to implement the cost-of-adjustment model outlined in Section 6. It also is the type of data required to distinguishbetween the productive and allocative efficiency effects of schooling and to fit demandfor health models in which medical care is not necessarily the primary health input. Fi-nally, it is the type of data to fully sort out the hypothesis that schooling causes healthfrom the competing hypothesis that time preference causes both using methods outlinedin Section 7.

These research efforts will not be easy, but their potential payoffs are substantial.Medical care markets in most countries are subject to large amounts of governmentintervention, regulation, and subsidization. I have emphasized the basic proposition thatconsumers demand health rather than medical care. Thus, one way to evaluate policyinitiatives aimed at medical care is to consider their impacts on health outcomes in thecontext of a cost-benefit analysis of programs that influence a variety of health inputs. Ifthis undertaking is to be successful, it must draw on refined estimates of the parametersof health production functions, output demand functions for health, and input demandfunctions for health inputs.

References

Angrist, J.D., and A.B. Krueger (1991), "Does compulsory school attendance affect schooling and earnings?",Quarterly Journal of Economics 106:979-1014.

Auster, R., I. Leveson and D. Sarachek (1969), "The production of health: an exploratory study", Journal ofHuman Resources 4:411-436.

Becker, G.S. (1964), Human Capital (Columbia University Press for the National Bureau of Economic Re-search, New York).

Becker, G.S. (1965), "A theory of the allocation of time", Economic Journal 75:493-517.Becker, G.S. (1967), Human Capital and the Personal Distribution of Income: An Analytical Approach (Uni-

versity of Michigan, Ann Arbor, MI). Also available in: G.S. Becker (1993), Human Capital, 3rd edn.(University of Chicago Press) 102-158.

Becker, G.S. (1996), Accounting for Tastes (Harvard University Press, Cambridge, MA).Becker, G.S., M. Grossman and K.M. Murphy (1991), "Rational addiction and the effect of price on con-

sumption", American Economic Review 81:237-241.Becker, G.S., M. Grossman and K.M. Murphy (1994), "An empirical analysis of cigarette addiction", Ameri-

can Economic Review 84:396-418.Becker, G.S., and C.B. Mulligan (1997), "The endogenous determination of time preference", Quarterly Jour-

nal of Economics 112:729-758.Becker, G.S., and K.M. Murphy (1988), "A theory of rational addiction", Journal of Political Economy

96:675-700.Ben-Porath, Y. (1967), "The production of human capital and the life cycle of earnings", Journal of Political

Economy 75:353-367.Bentham, J. (1931), Principles of Legislation (Harcourt, Brace and Co., New York).Berger, M.C., and J.P. Leigh (1989), "Schooling, self-selection, and health", Journal of Human Resources

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correlation between the instruments and the endogenous explanatory variable is weak", Journal of theAmerican Statistical Association 90:443-450.

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